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Nature 614 (7949), 676-681, 2023

Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction, offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error …

Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong, Alexander M Dalzell, Ashutosh Kumar, Phillip Helms, Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, Michael Kastoryano, Ryan Babbush, John Preskill, David R Reichman, Earl T Campbell, Edward F Valeev, Lin Lin, Garnet Kin-Lic Chan · Nature Communications 14 (1), 1952, 2023

Due to intense interest in the potential applications of quantum computing, it is critical to understand the basis for potential exponential quantum advantage in quantum chemistry. Here we gather the evidence for this case in the most common task in quantum chemistry, namely, ground-state energy estimation, for generic chemical problems where heuristic quantum state preparation might be assumed to be efficient. The availability of exponential quantum advantage then centers on whether features of the physical problem that enable efficient heuristic quantum state preparation also enable efficient solution by classical heuristics. Through numerical studies of quantum state preparation and empirical complexity analysis (including the error scaling) of classical heuristics, in both ab initio and model Hamiltonian settings, we conclude that evidence for such an exponential advantage across chemical space has yet to …

A Morvan, B Villalonga, X Mi, S Mandra, A Bengtsson, PV Klimov, Z Chen, S Hong, C Erickson, IK Drozdov, J Chau, G Laun, R Movassagh, A Asfaw, LTAN Brandão, R Peralta, D Abanin, R Acharya, R Allen, TI Andersen, K Anderson, M Ansmann, F Arute, K Arya, J Atalaya, JC Bardin, A Bilmes, G Bortoli, A Bourassa, J Bovaird, L Brill, M Broughton, BB Buckley, DA Buell, T Burger, B Burkett, N Bushnell, J Campero, HS Chang, B Chiaro, D Chik, C Chou, J Cogan, R Collins, P Conner, W Courtney, AL Crook, B Curtin, DM Debroy, A Barba, S Demura, A Di Paolo, A Dunsworth, L Faoro, E Farhi, R Fatemi, VS Ferreira, L Flores Burgos, E Forati, AG Fowler, B Foxen, G Garcia, E Genois, W Giang, C Gidney, D Gilboa, M Giustina, R Gosula, A Grajales Dau, JA Gross, S Habegger, MC Hamilton, M Hansen, MP Harrigan, SD Harrington, P Heu, MR Hoffmann, T Huang, A Huff, WJ Huggins, LB Ioffe, SV Isakov, J Iveland, E Jeffrey, Z Jiang, C Jones, P Juhas, D Kafri, T Khattar, M Khezri, M Kieferová, S Kim, A Kitaev, AR Klots, AN Korotkov, F Kostritsa, JM Kreikebaum, D Landhuis, P Laptev, K-M Lau, L Laws, J Lee, KW Lee, YD Lensky, BJ Lester, AT Lill, W Liu, A Locharla, FD Malone, O Martin, S Martin, JR McClean, M McEwen, KC Miao, A Mieszala, S Montazeri, W Mruczkiewicz, O Naaman, M Neeley, C Neill, A Nersisyan, M Newman, JH Ng, A Nguyen, M Nguyen, M Yuezhen Niu, TE O'Brien, S Omonije, A Opremcak, A Petukhov, R Potter, LP Pryadko, C Quintana, DM Rhodes, C Rocque, P Roushan, NC Rubin, N Saei, D Sank, K Sankaragomathi, KJ Satzinger, HF Schurkus, C Schuster, MJ Shearn, A Shorter, N Shutty, V Shvarts, V Sivak, J Skruzny, WC Smith · arXiv preprint arXiv:2304.11119, 2023

Quantum computers hold the promise of executing tasks beyond the capability of classical computers. Noise competes with coherent evolution and destroys long-range correlations, making it an outstanding challenge to fully leverage the computation power of near-term quantum processors. We report Random Circuit Sampling (RCS) experiments where we identify distinct phases driven by the interplay between quantum dynamics and noise. Using cross-entropy benchmarking, we observe phase boundaries which can define the computational complexity of noisy quantum evolution. We conclude by presenting an RCS experiment with 70 qubits at 24 cycles. We estimate the computational cost against improved classical methods and demonstrate that our experiment is beyond the capabilities of existing classical supercomputers.

Alexander Zlokapa, Benjamin Villalonga, Sergio Boixo, Daniel A Lidar · npj Quantum Information 9 (1), 36, 2023

Google’s quantum supremacy experiment heralded a transition point where quantum computers can evaluate a computational task, random circuit sampling, faster than classical supercomputers. We examine the constraints on the region of quantum advantage for quantum circuits with a larger number of qubits and gates than experimentally implemented. At near-term gate fidelities, we demonstrate that quantum supremacy is limited to circuits with a qubit count and circuit depth of a few hundred. Larger circuits encounter two distinct boundaries: a return of a classical advantage and practically infeasible quantum runtimes. Decreasing error rates cause the region of a quantum advantage to grow rapidly. At error rates required for early implementations of the surface code, the largest circuit size within the quantum supremacy regime coincides approximately with the smallest circuit size needed to implement error …

Jesse C Hoke, Matteo Ippoliti, Dmitry Abanin, Rajeev Acharya, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Joseph C Bardin, Andreas Bengtsson, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B Buckley, David A Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Zijun Chen, Ben Chiaro, Desmond Chik, Charina Chou, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L Crook, Ben Curtin, Alejandro Grajales Dau, Dripto M Debroy, Alexander Del Toro Barba, Sean Demura, Augustin Di Paolo, Ilya K Drozdov, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatem, Vinicius S Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Raja Gosula, Jonathan A Gross, Steve Habegger, Michael C Hamilton, Monica Hansen, Matthew P Harrigan, Sean D Harrington, Paula Heu, Markus R Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William J Huggins, Sergei V Isakov, Justin Iveland, Evan Jeffr, Cody Jones, Pavol Juhas, Dvir Kafri, Kostyantyn Kechedzhi, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V Klimov, Andrey R Klots, Alexander N Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny W Lee, Yuri D Lensky, Brian J Lester, Alexander T Lill, Wayne Liu, Aditya Locharla, Fionn D Malone, Orion Martin, Jarrod R McClean, Trevor McCourt, Matt McEwen, Kevin C Miao, Amanda Mieszala, Shirin Montazeri, Alexis Morvan, Ramis Movassagh, Wojciech Mruczkiewicz, Matthew Neeley, Charles Neill, Ani Nersisyan, Michael Newman, Jiun H Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Tom E O'Brien, Seun Omonije, Alex Opremcak, Andre Petukhov, Rebecca Potter, Leonid P Pryadko, Chris Quintana, Charles Rocque, Nicholas C Rubin, Negar Saei Daniel Sank, Kannan Sankaragomathi, Kevin J Satzinger, Henry F Schurkus, Christopher Schuster, Michael J Shearn, Aaron Shorter, Noah Shutty, Vlad Shvarts, Jindra Skruzny, W Clarke Smith, Rolando D Sterling, Douglas Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Ted White, Bryan WK Woo, Cheng Xing, Z Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, Nicholas Zobrist · arXiv preprint arXiv:2303.04792, 2023

Measurement has a special role in quantum theory: by collapsing the wavefunction it can enable phenomena such as teleportation and thereby alter the "arrow of time" that constrains unitary evolution. When integrated in many-body dynamics, measurements can lead to emergent patterns of quantum information in space-time that go beyond established paradigms for characterizing phases, either in or out of equilibrium. On present-day NISQ processors, the experimental realization of this physics is challenging due to noise, hardware limitations, and the stochastic nature of quantum measurement. Here we address each of these experimental challenges and investigate measurement-induced quantum information phases on up to 70 superconducting qubits. By leveraging the interchangeability of space and time, we use a duality mapping, to avoid mid-circuit measurement and access different manifestations of the underlying phases -- from entanglement scaling to measurement-induced teleportation -- in a unified way. We obtain finite-size signatures of a phase transition with a decoding protocol that correlates the experimental measurement record with classical simulation data. The phases display sharply different sensitivity to noise, which we exploit to turn an inherent hardware limitation into a useful diagnostic. Our work demonstrates an approach to realize measurement-induced physics at scales that are at the limits of current NISQ processors.

Oscar Higgott, Craig Gidney · arXiv preprint arXiv:2303.15933, 2023

In this work, we introduce a fast implementation of the minimum-weight perfect matching (MWPM) decoder, the most widely used decoder for several important families of quantum error correcting codes, including surface codes. Our algorithm, which we call sparse blossom, is a variant of the blossom algorithm which directly solves the decoding problem relevant to quantum error correction. Sparse blossom avoids the need for all-to-all Dijkstra searches, common amongst MWPM decoder implementations. For 0.1% circuit-level depolarising noise, sparse blossom processes syndrome data in both and bases of distance-17 surface code circuits in less than one microsecond per round of syndrome extraction on a single core, which matches the rate at which syndrome data is generated by superconducting quantum computers. Our implementation is open-source, and has been released in version 2 of the PyMatching library.

Matt McEwen, Dave Bacon, Craig Gidney · arXiv preprint arXiv:2302.02192, 2023

The typical time-independent view of quantum error correction (QEC) codes hides significant freedom in the decomposition into circuits that are executable on hardware. Using the concept of detecting regions, we design time-dynamic QEC circuits directly instead of designing static QEC codes to decompose into circuits. In particular, we improve on the standard circuit constructions for the surface code, presenting new circuits that can embed on a hexagonal grid instead of a square grid, that can use ISWAP gates instead of CNOT or CZ gates, that can exchange qubit data and measure roles, and that move logical patches around the physical qubit grid while executing. All these constructions use no additional entangling gate layers and display essentially the same logical performance, having teraquop footprints within 25% of the standard surface code circuit. We expect these circuits to be of great interest to quantum hardware engineers, because they achieve essentially the same logical performance as standard surface code circuits while relaxing demands on hardware.

Theodore White, Alex Opremcak, George Sterling, Alexander Korotkov, Daniel Sank, Rajeev Acharya, Markus Ansmann, Frank Arute, Kunal Arya, Joseph C Bardin, Andreas Bengtsson, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Bob B Buckley, David A Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Zijun Chen, Ben Chiaro, Josh Cogan, Roberto Collins, Alexander L Crook, Ben Curtin, Sean Demura, Andrew Dunsworth, Catherine Erickson, Reza Fatemi, Leslie Flores Burgos, Ebrahim Forati, Brooks Foxen, William Giang, Marissa Giustina, Alejandro Grajales Dau, Michael C Hamilton, Sean D Harrington, Jeremy Hilton, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, Justin Iveland, Evan Jeffrey, Mária Kieferová, Seon Kim, Paul V Klimov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Lily Laws, Kenny Lee, Brian J Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Trevor McCourt, Matt McEwen, Xiao Mi, Kevin C Miao, Shirin Montazeri, Alexis Morvan, Matthew Neeley, Charles Neill, Ani Nersisyan, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Rebecca Potter, Chris Quintana, Pedram Roushan, Kannan Sankaragomathi, Kevin J Satzinger, Christopher Schuster, Michael J Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, W Clarke Smith, Marco Szalay, Alfredo Torres, Bryan WK Woo, Z Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Ningfeng Zhu, Nicholas Zobrist, Yu Chen, Anthony Megrant, Julian Kelly, Ofer Naaman · Applied Physics Letters 122 (1), 2023

We demonstrate a high dynamic range Josephson parametric amplifier (JPA) in which the active nonlinear element is implemented using an array of rf-SQUIDs. The device is matched to the 50 Ω environment with a Klopfenstein-taper impedance transformer and achieves a bandwidth of 250–300 MHz with input saturation powers up to− 95 dBm at 20 dB gain. A 54-qubit Sycamore processor was used to benchmark these devices, providing a calibration for readout power, an estimation of amplifier added noise, and a platform for comparison against standard impedance matched parametric amplifiers with a single dc-SQUID. We find that the high power rf-SQUID array design has no adverse effect on system noise, readout fidelity, or qubit dephasing, and we estimate an upper bound on amplifier added noise at 1.6 times the quantum limit. Finally, amplifiers with this design show no degradation in readout fidelity due …

Craig Gidney · Quantum 7, 1156, 2023

In this paper, I present a way to compile the surface code into two-body parity measurements (" pair measurements"), where the pair measurements run along the edges of a Cairo pentagonal tiling. The resulting circuit improves on prior work by Chao et al. by using fewer pair measurements per four-body stabilizer measurement (5 instead of 6) and fewer time steps per round of stabilizer measurement (6 instead of 10). Using Monte Carlo sampling, I show that these improvements increase the threshold of the surface code when compiling into pair measurements from to , and also that they improve the teraquop footprint at a physical gate error rate from qubits to qubits. However, I also show that the teraquop footprint of Chao et al's construction improves more quickly than mine as physical error rate decreases, and is likely better below a physical gate error rate of (due to bidirectional hook errors in my construction). I also compare to the planar honeycomb code, showing that although this work does noticeably reduce the gap between the surface code and the honeycomb code (when compiling into pair measurements), the honeycomb code is still more efficient (threshold , teraquop footprint at of ).

Nicholas C Rubin, Dominic W Berry, Fionn D Malone, Alec F White, Tanuj Khattar, A Eugene DePrince III, Sabrina Sicolo, Michael Küehn, Michael Kaicher, Joonho Lee, Ryan Babbush · PRX Quantum 4 (4), 040303, 2023

The simulation of chemistry is among the most promising applications of quantum computing. However, most prior work exploring algorithms for block encoding, time evolving, and sampling in the eigenbasis of electronic structure Hamiltonians has either focused on modeling finite-sized systems, or has required a large number of plane-wave basis functions. In this work, we extend methods for quantum simulation with Bloch orbitals constructed from symmetry-adapted atom-centered orbitals so that one can model periodic ab initio Hamiltonians using only a modest number of basis functions. We focus on adapting existing algorithms based on combining qubitization with tensor factorizations of the Coulomb operator. Significant modifications of those algorithms are required to obtain an asymptotic speedup leveraging translational (or, more broadly, Abelian) symmetries. We implement block encodings using known …

Amira Abbas, Robbie King, Hsin-Yuan Huang, William J Huggins, Ramis Movassagh, Dar Gilboa, Jarrod R McClean · arXiv preprint arXiv:2305.13362, 2023

The success of modern deep learning hinges on the ability to train neural networks at scale. Through clever reuse of intermediate information, backpropagation facilitates training through gradient computation at a total cost roughly proportional to running the function, rather than incurring an additional factor proportional to the number of parameters - which can now be in the trillions. Naively, one expects that quantum measurement collapse entirely rules out the reuse of quantum information as in backpropagation. But recent developments in shadow tomography, which assumes access to multiple copies of a quantum state, have challenged that notion. Here, we investigate whether parameterized quantum models can train as efficiently as classical neural networks. We show that achieving backpropagation scaling is impossible without access to multiple copies of a state. With this added ability, we introduce an algorithm with foundations in shadow tomography that matches backpropagation scaling in quantum resources while reducing classical auxiliary computational costs to open problems in shadow tomography. These results highlight the nuance of reusing quantum information for practical purposes and clarify the unique difficulties in training large quantum models, which could alter the course of quantum machine learning.

Ryan Babbush, Dominic W Berry, Robin Kothari, Rolando D Somma, Nathan Wiebe · arXiv preprint arXiv:2303.13012, 2023

We present a quantum algorithm for simulating the classical dynamics of coupled oscillators (e.g., masses coupled by springs). Our approach leverages a mapping between the Schr\"odinger equation and Newton's equations for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators. When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in , almost linear in the evolution time, and sublinear in the sparsity. As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time, for a specification of the problem that we prove is BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers. Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with modes.

X Mi, AA Michailidis, S Shabani, KC Miao, PV Klimov, J Lloyd, E Rosenberg, R Acharya, I Aleiner, TI Andersen, M Ansmann, F Arute, K Arya, A Asfaw, J Atalaya, JC Bardin, A Bengtsson, G Bortoli, A Bourassa, J Bovaird, L Brill, M Broughton, BB Buckley, DA Buell, T Burger, B Burkett, N Bushnell, Z Chen, B Chiaro, D Chik, C Chou, J Cogan, R Collins, P Conner, W Courtney, AL Crook, B Curtin, AG Dau, DM Debroy, A Barba, S Demura, A Di Paolo, IK Drozdov, A Dunsworth, C Erickson, L Faoro, E Farhi, R Fatemi, VS Ferreira, LF Forati, AG Fowler, B Foxen, E Genois, W Giang, C Gidney, D Gilboa, M Giustina, R Gosula, JA Gross, S Habegger, MC Hamilton, M Hansen, MP Harrigan, SD Harrington, P Heu, MR Hoffmann, S Hong, T Huang, A Huff, WJ Huggins, LB Ioffe, SV Isakov, J Iveland, E Jeffrey, Z Jiang, C Jones, P Juhas, D Kafri, K Kechedzhi, T Khattar, M Khezri, M Kieferova, S Kim, A Kitaev, AR Klots, AN Korotkov, F Kostritsa, JM Kreikebaum, D Landhuis, P Laptev, K-M Lau, L Laws, J Lee, KW Lee, YD Lensky, BJ Lester, AT Lill, W Liu, A Locharla, FD Malone, O Martin, JR McClean, M McEwen, A Mieszala, S Montazeri, A Morvan, R Movassagh, W Mruczkiewicz, M Neeley, C Neill, A Nersisyan, M Newman, JH Ng, A Nguyen, M Nguyen, MY Niu, TE OBrien, A Opremcak, A Petukhov, R Potter, LP Pryadko, C Quintana, C Rocque, NC Rubin, N Saei, D Sank, K Sankaragomathi, KJ Satzinger, HF Schurkus, C Schuster, MJ Shearn, A Shorter, N Shutty, V Shvarts, J Skruzny, WC Smith, R Somma, G Sterling, D Strain, M Szalay, A Torres, G Vidal, B Villalonga, CV Heidweiller, T White, BWK Woo, C Xing, ZJ Yao, P Yeh, J Yoo · arXiv preprint arXiv:2304.13878, 2023

Engineered dissipative reservoirs can steer many-body quantum systems toward correlated steady states useful for quantum simulation of high-temperature superconductivity or quantum magnetism. Using 49 superconducting qubits, we prepare low-energy states of the transverse-field Ising model via coupling to dissipative auxiliary qubits. In 1D, we observe long-range quantum correlations and a ground state fidelity that depends weakly on system sizes. In 2D, we find mutual information that extends beyond nearest neighbors. Lastly, by coupling the system to auxiliaries emulating reservoirs with different chemical potentials, we discover a new spin transport regime in the quantum Heisenberg model. Our results establish engineered dissipation as a scalable alternative to unitary evolution for preparing entangled many-body states on noisy quantum processors, and an essential tool for investigating nonequilibrium quantum phenomena.

Randy Kwende, Theodore White, Ofer Naaman · Applied Physics Letters 122 (22), 2023

We demonstrate a 3-port Josephson parametric circulator matched to 50X using second order Chebyshev networks. The device notably operates with two of its signal ports at the same frequency and uses only two out-of-phase pumps at a single frequency. As a consequence, when operated as an isolator, it does not require phase coherence between the pumps and the signal, thus simplifying the requirements for its integration into standard dispersive qubit readout setups. The device utilizes parametric couplers based on a balanced bridge of rf-superconducting quantum interference device arrays, which offer purely parametric coupling and high dynamic range. We characterize the device by measuring its full 3 Â3 S-matrix as a function of frequency and the relative phase between the two pumps. We find up to 15dB nonreciprocity over a 200 MHz signal band, port match better than 10 dB, low insertion loss of 0.6 …

Raffaele Santagati, Alan Aspuru-Guzik, Ryan Babbush, Matthias Degroote, Leticia Gonzalez, Elica Kyoseva, Nikolaj Moll, Markus Oppel, Robert M Parrish, Nicholas C Rubin, Michael Streif, Christofer S Tautermann, Horst Weiss, Nathan Wiebe, Clemens Utschig-Utschig · arXiv preprint arXiv:2301.04114, 2023

Quantum computers promise to impact industrial applications, for which quantum chemical calculations are required, by virtue of their high accuracy. This perspective explores the challenges and opportunities of applying quantum computers to drug design, discusses where they could transform industrial research and elaborates on what is needed to reach this goal.

Ryan Babbush, William J Huggins, Dominic W Berry, Shu Fay Ung, Andrew Zhao, David R Reichman, Hartmut Neven, Andrew D Baczewski, Joonho Lee · Nature Communications 14, 4058, 2023

Quantum algorithms for simulating electronic ground states are slower than popular classical mean-field algorithms such as Hartree-Fock and density functional theory, but offer higher accuracy. Accordingly, quantum computers have been predominantly regarded as competitors to only the most accurate and costly classical methods for treating electron correlation. However, here we tighten bounds showing that certain first quantized quantum algorithms enable exact time evolution of electronic systems with exponentially less space and polynomially fewer operations in basis set size than conventional real-time time-dependent Hartree-Fock and density functional theory. Although the need to sample observables in the quantum algorithm reduces the speedup, we show that one can estimate all elements of the -particle reduced density matrix with a number of samples scaling only polylogarithmically in basis set size. We also introduce a more efficient quantum algorithm for first quantized mean-field state preparation that is likely cheaper than the cost of time evolution. We conclude that quantum speedup is most pronounced for finite temperature simulations and suggest several practically important electron dynamics problems with potential quantum advantage.

Eliott Rosenberg, Trond Andersen, Rhine Samajdar, Andre Petukhov, Jesse Hoke, Dmitry Abanin, Andreas Bengtsson, Ilya Drozdov, Catherine Erickson, Paul Klimov, Xiao Mi, Alexis Morvan, Matthew Neeley, Charles Neill, Rajeev Acharya, Igor Aleiner, Richard Allen, Kyle Anderson, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Joseph Bardin, A Bilmes, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B Buckley, David Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Juan Campero, Hung-Shen Chang, Zijun Chen, Benjamin Chiaro, Desmond Chik, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander Crook, Ben Curtin, Dripto Debroy, Alexander Del Toro Barba, Sean Demura, Agustin Di Paolo, Andrew Dunsworth, Clint Earle, E Farhi, Reza Fatemi, Vinicius Ferreira, Leslie Flores, Ebrahim Forati, Austin Fowler, Brooks Foxen, Gonzalo Garcia, Élie Genois, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Jonathan Gross, Steve Habegger, Michael Hamilton, Monica Hansen, Matthew Harrigan, Sean Harrington, Paula Heu, Gordon Hill, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William Huggins, Lev Ioffe, Sergei Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, D Kafri, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Andrey Klots, Alexander Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim Ming Lau, Lily Laws, Joonho Lee, Kenneth Lee, Yuri Lensky, Brian Lester, Alexander Lill, Wayne Liu, William P Livingston, A Locharla, Salvatore Mandrà, Orion Martin, Steven Martin, Jarrod McClean, Matthew McEwen, Seneca Meeks, Kevin Miao, Amanda Mieszala, Shirin Montazeri, Ramis Movassagh, Wojciech Mruczkiewicz, Ani Nersisyan, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, M Niu, Thomas O'Brien, Seun Omonije, Alex Opremcak, Rebecca Potter, Leonid Pryadko, Chris Quintana, David Rhodes, Charles Rocque, N Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin Satzinger, Henry Schurkus, Christopher Schuster, Michael Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Volodymyr Sivak, Jindra Skruzny, Clarke Smith, Rolando Somma, George Sterling · arXiv preprint arXiv:2306.09333, 2023

Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the 1D Heisenberg model were conjectured to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we study the probability distribution, , of the magnetization transferred across the chain's center. The first two moments of show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments rule out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide key insights into universal behavior in quantum systems.

Dar Gilboa, Jarrod R McClean · arXiv preprint arXiv:2310.07136, 2023

Training and inference with large machine learning models that far exceed the memory capacity of individual devices necessitates the design of distributed architectures, forcing one to contend with communication constraints. We present a framework for distributed computation over a quantum network in which data is encoded into specialized quantum states. We prove that for certain models within this framework, inference and training using gradient descent can be performed with exponentially less communication compared to their classical analogs, and with relatively modest time and space complexity overheads relative to standard gradient-based methods. To our knowledge, this is the first example of exponential quantum advantage for a generic class of machine learning problems with dense classical data that holds regardless of the data encoding cost. Moreover, we show that models in this class can encode highly nonlinear features of their inputs, and their expressivity increases exponentially with model depth. We also find that, interestingly, the communication advantage nearly vanishes for simpler linear classifiers. These results can be combined with natural privacy advantages in the communicated quantum states that limit the amount of information that can be extracted from them about the data and model parameters. Taken as a whole, these findings form a promising foundation for distributed machine learning over quantum networks.

Yilun Yang, Arthur Christianen, Sandra Coll-Vinent, Vadim Smelyanskiy, Mari Carmen Bañuls, Thomas E O’Brien, Dominik S Wild, J Ignacio Cirac · PRX Quantum 4 (3), 030320, 2023

Quantum simulation is one of the most promising scientific applications of quantum computers. Due to decoherence and noise in current devices, it is however challenging to perform digital quantum simulation in a regime that is intractable with classical computers. In this work, we propose an experimental protocol for probing dynamics and equilibrium properties on near-term digital quantum computers. As a key ingredient of our work, we show that it is possible to study thermalization even with a relatively coarse Trotter decomposition of the Hamiltonian evolution of interest. Even though the step size is too large to permit a rigorous bound on the Trotter error, we observe that the system prethermalizes in accordance with previous results for Floquet systems. The dynamics closely resemble the thermalization of the model underlying the Trotterization up to long times. We make our approach resilient to noise by …

Andreas Bengtsson, Alex Opremcak, Mostafa Khezri, Daniel Sank, Alexandre Bourassa, Kevin J Satzinger, Sabrina Hong, Catherine Erickson, Brian J Lester, Kevin C Miao, Alexander N Korotkov, Julian Kelly, Zijun Chen, Paul V Klimov · arXiv preprint arXiv:2308.02079, 2023

Measurement is an essential component of quantum algorithms, and for superconducting qubits it is often the most error prone. Here, we demonstrate model-based readout optimization achieving low measurement errors while avoiding detrimental side-effects. For simultaneous and mid-circuit measurements across 17 qubits, we observe 1.5% error per qubit with a 500ns end-to-end duration and minimal excess reset error from residual resonator photons. We also suppress measurement-induced state transitions achieving a leakage rate limited by natural heating. This technique can scale to hundreds of qubits and be used to enhance the performance of error-correcting codes and near-term applications.

Ryan Kaufman, Theodore White, Mark I Dykman, Andrea Iorio, George Stirling, Sabrina Hong, Alex Opremcak, Andreas Bengtsson, Lara Faoro, Joseph C Bardin, Tim Burger, Robert Gasca, Ofer Naaman · arXiv preprint arXiv:2305.17816, 2023

We demonstrate a Josephson parametric amplifier design with a band-pass impedance matching network based on a third-order Chebyshev prototype. We measured eight amplifiers operating at 4.6 GHz that exhibit gains of 20 dB with less than 1 dB gain ripple and up to 500 MHz bandwidth. The amplifiers further achieve high output saturation powers around -73 dBm based on the use of rf-SQUID arrays as their nonlinear element. We characterize the system readout efficiency and its signal-to-noise ratio near saturation using a Sycamore processor, finding the data consistent with near quantum limited noise performance of the amplifiers. In addition, we measure the amplifiers' intermodulation distortion in two-tone experiments as a function of input power and inter-tone detuning, and observe excess distortion at small detuning with a pronounced dip as a function of signal power, which we interpret in terms of power-dependent dielectric losses.

Paul V Klimov, Andreas Bengtsson, Chris Quintana, Alexandre Bourassa, Sabrina Hong, Andrew Dunsworth, Kevin J Satzinger, William P Livingston, Volodymyr Sivak, Murphy Y Niu, Trond I Andersen, Yaxing Zhang, Desmond Chik, Zijun Chen, Charles Neill, Catherine Erickson, Alejandro Grajales Dau, Anthony Megrant, Pedram Roushan, Alexander N Korotkov, Julian Kelly, Vadim Smelyanskiy, Yu Chen, Hartmut Neven · arXiv preprint arXiv:2308.02321, 2023

A foundational assumption of quantum error correction theory is that quantum gates can be scaled to large processors without exceeding the error-threshold for fault tolerance. Two major challenges that could become fundamental roadblocks are manufacturing high performance quantum hardware and engineering a control system that can reach its performance limits. The control challenge of scaling quantum gates from small to large processors without degrading performance often maps to non-convex, high-constraint, and time-dependent control optimization over an exponentially expanding configuration space. Here we report on a control optimization strategy that can scalably overcome the complexity of such problems. We demonstrate it by choreographing the frequency trajectories of 68 frequency-tunable superconducting qubits to execute single- and two-qubit gates while mitigating computational errors. When combined with a comprehensive model of physical errors across our processor, the strategy suppresses physical error rates by compared with the case of no optimization. Furthermore, it is projected to achieve a similar performance advantage on a distance-23 surface code logical qubit with 1057 physical qubits. Our control optimization strategy solves a generic scaling challenge in a way that can be adapted to other quantum algorithms, operations, and computing architectures.

Sofia Gonzalez-Garcia, Shengqi Sang, Timothy H Hsieh, Sergio Boixo, Guifre Vidal, Andrew C Potter, Romain Vasseur · arXiv preprint arXiv:2307.11053, 2023

Projected entangled pair states (PEPS) offer memory-efficient representations of some quantum many-body states that obey an entanglement area law, and are the basis for classical simulations of ground states in two-dimensional (2d) condensed matter systems. However, rigorous results show that exactly computing observables from a 2d PEPS state is generically a computationally hard problem. Yet approximation schemes for computing properties of 2d PEPS are regularly used, and empirically seen to succeed, for a large subclass of (not too entangled) condensed matter ground states. Adopting the philosophy of random matrix theory, in this work we analyze the complexity of approximately contracting a 2d random PEPS by exploiting an analytic mapping to an effective replicated statistical mechanics model that permits a controlled analysis at large bond dimension. Through this statistical-mechanics lens, we argue that: i) although approximately sampling wave-function amplitudes of random PEPS faces a computational-complexity phase transition above a critical bond dimension, ii) one can generically efficiently estimate the norm and correlation functions for any finite bond dimension. These results are supported numerically for various bond-dimension regimes. It is an important open question whether the above results for random PEPS apply more generally also to PEPS representing physically relevant ground states

Craig Gidney, Dave Bacon · arXiv preprint arXiv:2305.12046, 2023

We give the Bacon-Shor code a threshold purely by deleting gates from its circuit. Specifically: we use lattice surgery to concatenate the Bacon-Shor code with itself using local planar connectivity, and observe that the resulting circuit is a subset of the circuit that would be used by a larger Bacon-Shor code.

William J Huggins, Jarrod R McClean · arXiv preprint arXiv:2305.09638, 2023

Real-world applications of computing can be extremely time-sensitive. It would be valuable if we could accelerate such tasks by performing some of the work ahead of time. Motivated by this, we propose a cost model for quantum algorithms that allows quantum precomputation; i.e., for a polynomial amount of "free" computation before the input to an algorithm is fully specified, and methods for taking advantage of it. We analyze two families of unitaries that are asymptotically more efficient to implement in this cost model than in the standard one. The first example of quantum precomputation, based on density matrix exponentiation, could offer an exponential advantage under certain conditions. The second example uses a variant of gate teleportation to achieve a quadratic advantage when compared with implementing the unitaries directly. These examples hint that quantum precomputation may offer a new arena in which to seek quantum advantage.

Ryan Kaufman, Ofer Naaman · arXiv preprint arXiv:2303.00184, 2023

In this note we describe Josephson parametric amplifier (JPA) matching networks based on Legendre polynomials. These networks typically exhibit lower ripple and gentler roll-off than Chebyshev networks with similar parameters, and can be viewed as bridging the gap between Butterworth and Chebyshev ones. We tabulate prototype coefficients for parametric amplifiers based on Legendre polynomials with a range of gain and ripple parameters, and for a range of network orders. We also use this opportunity to further illustrate the synthesis of these networks based on methods from previous work, and synthesize a prototype JPA with 20dB gain at a center frequency of 5GHz with a bandwidth of 500MHz.

Craig Gidney · arXiv preprint arXiv:2302.12292, 2023

In this paper, I show how an intentional hook error mechanism can be used as a control knob for injecting magic states into surface codes. The limitation, and benefit, of this approach is that it can only inject states in the XY or YZ plane of the Bloch sphere. This increases fidelity, because perturbations out of the target plane can be detected as errors. I use Monte Carlo sampling to show that this technique outperforms previous injection techniques, achieving lower error rates at smaller spacetime cost under digitized circuit noise.

Kianna Wan, William J Huggins, Joonho Lee, Ryan Babbush · Communications in Mathematical Physics, 1-72, 2023

"Classical shadows" are estimators of an unknown quantum state, constructed from suitably distributed random measurements on copies of that state [Nature Physics 16, 1050-1057]. Here, we analyze classical shadows obtained using random matchgate circuits, which correspond to fermionic Gaussian unitaries. We prove that the first three moments of the Haar distribution over the continuous group of matchgate circuits are equal to those of the discrete uniform distribution over only the matchgate circuits that are also Clifford unitaries; thus, the latter forms a "matchgate 3-design." This implies that the classical shadows resulting from the two ensembles are functionally equivalent. We show how one can use these matchgate shadows to efficiently estimate inner products between an arbitrary quantum state and fermionic Gaussian states, as well as the expectation values of local fermionic operators and various other quantities, thus surpassing the capabilities of prior work. As a concrete application, this enables us to apply wavefunction constraints that control the fermion sign problem in the quantum-classical auxiliary-field quantum Monte Carlo algorithm (QC-AFQMC) [Nature 603, 416-420], without the exponential post-processing cost incurred by the original approach.

Mostafa Khezri, Alex Opremcak, Zijun Chen, Kevin C Miao, Matt McEwen, Andreas Bengtsson, Theodore White, Ofer Naaman, Daniel Sank, Alexander N Korotkov, Yu Chen, Vadim Smelyanskiy · Physical Review Applied 20 (5), 054008, 2023

Superconducting qubits typically use a dispersive readout scheme, where a resonator is coupled to a qubit such that its frequency is qubit-state dependent. Measurement is performed by driving the resonator, where the transmitted resonator field yields information about the resonator frequency and thus the qubit state. Ideally, we could use arbitrarily strong resonator drives to achieve a target signal-to-noise ratio in the shortest possible time. However, experiments have shown that when the average resonator photon number exceeds a certain threshold, the qubit is excited out of its computational subspace in a process we refer to as a measurement-induced state transition (MIST). These transitions degrade readout fidelity, and constitute leakage, which precludes further operation of the qubit in, for example, error correction. Here we study these transitions experimentally with a transmon qubit by measuring their …

Hsin-Yuan Huang, Michael Broughton, Jordan Cotler, Sitan Chen, Jerry Li, Masoud Mohseni, Hartmut Neven, Ryan Babbush, Richard Kueng, John Preskill, Jarrod R McClean · Science 376 (6598), 1182-1186, 2022

Quantum technology promises to revolutionize how we learn about the physical world. An experiment that processes quantum data with a quantum computer could have substantial advantages over conventional experiments in which quantum states are measured and outcomes are processed with a classical computer. We proved that quantum machines could learn from exponentially fewer experiments than the number required by conventional experiments. This exponential advantage is shown for predicting properties of physical systems, performing quantum principal component analysis, and learning about physical dynamics. Furthermore, the quantum resources needed for achieving an exponential advantage are quite modest in some cases. Conducting experiments with 40 superconducting qubits and 1300 quantum gates, we demonstrated that a substantial quantum advantage is possible with today’s …

Xiao Mi, Matteo Ippoliti, Chris Quintana, Ami Greene, Zijun Chen, Jonathan Gross, Frank Arute, Kunal Arya, Juan Atalaya, Ryan Babbush, Joseph C Bardin, Joao Basso, Andreas Bengtsson, Alexander Bilmes, Alexandre Bourassa, Leon Brill, Michael Broughton, Bob B Buckley, David A Buell, Brian Burkett, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, William Courtney, Dripto Debroy, Sean Demura, Alan R Derk, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Edward Farhi, Austin G Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Matthew P Harrigan, Sean D Harrington, Jeremy Hilton, Alan Ho, Sabrina Hong, Trent Huang, Ashley Huff, William J Huggins, LB Ioffe, Sergei V Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Tanuj Khattar, Seon Kim, Alexei Kitaev, Paul V Klimov, Alexander N Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Joonho Lee, Kenny Lee, Aditya Locharla, Erik Lucero, Orion Martin, Jarrod R McClean, Trevor McCourt, Matt McEwen, Kevin C Miao, Masoud Mohseni, Shirin Montazeri, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Michael Newman, Murphy Yuezhen Niu, Thomas E O’Brien, Alex Opremcak, Eric Ostby, Balint Pato, Andre Petukhov, Nicholas C Rubin, Daniel Sank, Kevin J Satzinger, Vladimir Shvarts, Yuan Su, Doug Strain, Marco Szalay, Matthew D Trevithick, Benjamin Villalonga, Theodore White, Z Jamie Yao, Ping Yeh, Juhwan Yoo, Adam Zalcman, Hartmut Neven, Sergio Boixo, Vadim Smelyanskiy, Anthony Megrant, Julian Kelly, Yu Chen, SL Sondhi, Roderich Moessner, Kostyantyn Kechedzhi, Vedika Khemani, Pedram Roushan · Nature 601 (7894), 531-536, 2022

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases– that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC),–. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order,,. In eigenstate-ordered MBL phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, or from regimes in which the dynamics of a few select states can mask typical behaviour …

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Leo Zhou · Quantum 6, 759, 2022

The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers . While QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can efficiently find an approximate solution for a typical instance of the SK model to within times the ground state energy. We hope to match its performance with the QAOA.

William J Huggins, Bryan A O’Gorman, Nicholas C Rubin, David R Reichman, Ryan Babbush, Joonho Lee · Nature 603 (7901), 416-420, 2022

Interacting many-electron problems pose some of the greatest computational challenges in science, with essential applications across many fields. The solutions to these problems will offer accurate predictions of chemical reactivity and kinetics, and other properties of quantum systems–. Fermionic quantum Monte Carlo (QMC) methods,, which use a statistical sampling of the ground state, are among the most powerful approaches to these problems. Controlling the fermionic sign problem with constraints ensures the efficiency of QMC at the expense of potentially significant biases owing to the limited flexibility of classical computation. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We implement our scheme experimentally using up to 16 qubits to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals …

Matt McEwen, Lara Faoro, Kunal Arya, Andrew Dunsworth, Trent Huang, Seon Kim, Brian Burkett, Austin Fowler, Frank Arute, Joseph C Bardin, Andreas Bengtsson, Alexander Bilmes, Bob B Buckley, Nicholas Bushnell, Zijun Chen, Roberto Collins, Sean Demura, Alan R Derk, Catherine Erickson, Marissa Giustina, Sean D Harrington, Sabrina Hong, Evan Jeffrey, Julian Kelly, Paul V Klimov, Fedor Kostritsa, Pavel Laptev, Aditya Locharla, Xiao Mi, Kevin C Miao, Shirin Montazeri, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Alex Opremcak, Chris Quintana, Nicholas Redd, Pedram Roushan, Daniel Sank, Kevin J Satzinger, Vladimir Shvarts, Theodore White, Z Jamie Yao, Ping Yeh, Juhwan Yoo, Yu Chen, Vadim Smelyanskiy, John M Martinis, Hartmut Neven, Anthony Megrant, Lev Ioffe, Rami Barends · Nature Physics 18 (1), 107-111, 2022

Scalable quantum computing can become a reality with error correction, provided that coherent qubits can be constructed in large arrays,. The key premise is that physical errors can remain both small and sufficiently uncorrelated as devices scale, so that logical error rates can be exponentially suppressed. However, impacts from cosmic rays and latent radioactivity violate these assumptions. An impinging particle can ionize the substrate and induce a burst of quasiparticles that destroys qubit coherence throughout the device. High-energy radiation has been identified as a source of error in pilot superconducting quantum devices–, but the effect on large-scale algorithms and error correction remains an open question. Elucidating the physics involved requires operating large numbers of qubits at the same rapid timescales necessary for error correction. Here, we use space- and time-resolved measurements of a …

Zhenyu Cai, Ryan Babbush, Simon C Benjamin, Suguru Endo, William J Huggins, Ying Li, Jarrod R McClean, Thomas E O'Brien · arXiv preprint arXiv:2210.00921, 2022

For quantum computers to successfully solve real-world problems, it is necessary to tackle the challenge of noise: the errors which occur in elementary physical components due to unwanted or imperfect interactions. The theory of quantum fault tolerance can provide an answer in the long term, but in the coming era of `NISQ' machines we must seek to mitigate errors rather than completely remove them. This review surveys the diverse methods that have been proposed for quantum error mitigation, assesses their in-principle efficacy, and then describes the hardware demonstrations achieved to date. We identify the commonalities and limitations among the methods, noting how mitigation methods can be chosen according to the primary type of noise present, including algorithmic errors. Open problems in the field are identified and we discuss the prospects for realising mitigation-based devices that can deliver quantum advantage with an impact on science and business.

Daniel Jafferis, Alexander Zlokapa, Joseph D Lykken, David K Kolchmeyer, Samantha I Davis, Nikolai Lauk, Hartmut Neven, Maria Spiropulu · Nature 612 (7938), 51-55, 2022

The holographic principle, theorized to be a property of quantum gravity, postulates that the description of a volume of space can be encoded on a lower-dimensional boundary. The anti-de Sitter (AdS)/conformal field theory correspondence or duality is the principal example of holography. The Sachdev–Ye–Kitaev (SYK) model of N ≫ 1 Majorana fermions, has features suggesting the existence of a gravitational dual in AdS2, and is a new realization of holography–. We invoke the holographic correspondence of the SYK many-body system and gravity to probe the conjectured ER=EPR relation between entanglement and spacetime geometry, through the traversable wormhole mechanism as implemented in the SYK model,. A qubit can be used to probe the SYK traversable wormhole dynamics through the corresponding teleportation protocol. This can be realized as a quantum circuit, equivalent to the …

Murphy Yuezhen Niu, Alexander Zlokapa, Michael Broughton, Sergio Boixo, Masoud Mohseni, Vadim Smelyanskyi, Hartmut Neven · Physical Review Letters 128 (22), 220505, 2022

Generative adversarial networks (GANs) are one of the most widely adopted machine learning methods for data generation. In this work, we propose a new type of architecture for quantum generative adversarial networks (an entangling quantum GAN, EQ-GAN) that overcomes limitations of previously proposed quantum GANs. Leveraging the entangling power of quantum circuits, the EQ-GAN converges to the Nash equilibrium by performing entangling operations between both the generator output and true quantum data. In the first multiqubit experimental demonstration of a fully quantum GAN with a provably optimal Nash equilibrium, we use the EQ-GAN on a Google Sycamore superconducting quantum processor to mitigate uncharacterized errors, and we numerically confirm successful error mitigation with simulations up to 18 qubits. Finally, we present an application of the EQ-GAN to prepare an approximate …

Ben Chiaro, C Neill, A Bohrdt, Michele Filippone, F Arute, K Arya, R Babbush, D Bacon, J Bardin, R Barends, S Boixo, D Buell, B Burkett, Y Chen, Z Chen, R Collins, A Dunsworth, E Farhi, A Fowler, B Foxen, C Gidney, M Giustina, M Harrigan, T Huang, S Isakov, E Jeffrey, Z Jiang, D Kafri, K Kechedzhi, J Kelly, P Klimov, A Korotkov, F Kostritsa, D Landhuis, E Lucero, J McClean, X Mi, A Megrant, M Mohseni, J Mutus, M McEwen, O Naaman, M Neeley, M Niu, A Petukhov, C Quintana, N Rubin, D Sank, K Satzinger, T White, Z Yao, P Yeh, A Zalcman, V Smelyanskiy, H Neven, S Gopalakrishnan, D Abanin, M Knap, J Martinis, P Roushan · Physical Review Research 4 (1), 013148, 2022

The interplay of interactions and strong disorder can lead to an exotic quantum many-body localized (MBL) phase of matter. Beyond the absence of transport, the MBL phase has distinctive signatures, such as slow dephasing and logarithmic entanglement growth; they commonly result in slow and subtle modifications of the dynamics, rendering their measurement challenging. Here, we experimentally characterize these properties of the MBL phase in a system of coupled superconducting qubits. By implementing phase sensitive techniques, we map out the structure of local integrals of motion in the MBL phase. Tomographic reconstruction of single and two-qubit density matrices allows us to determine the spatial and temporal entanglement growth between the localized sites. In addition, we study the preservation of entanglement in the MBL phase. The interferometric protocols implemented here detect affirmative …

Pedro CS Costa, Dong An, Yuval R Sanders, Yuan Su, Ryan Babbush, Dominic W Berry · PRX Quantum 3 (4), 040303, 2022

Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. Such approaches have enabled near-linear scaling in the condition number κ of the linear system, without requiring a complicated variable-time amplitude amplification procedure. However, the most efficient of those procedures is still asymptotically suboptimal by a factor of log(κ). Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete-time evolutions. In combination with the qubitized quantum walk, our discrete adiabatic theorem gives a speed-up for all adiabatic algorithms. Here, we use this combination to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in κ, matching a …

Yu Tong, Victor V Albert, Jarrod R McClean, John Preskill, Yuan Su · Quantum 6, 816, 2022

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit on each local quantum number, and if the initial state has low local quantum numbers, then an error at most can be achieved by choosing to scale polylogarithmically with , an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on that achieves accuracy , obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error …

Xiao Mi, M Sonner, M Yuezhen Niu, KW Lee, B Foxen, R Acharya, I Aleiner, TI Andersen, F Arute, K Arya, A Asfaw, J Atalaya, JC Bardin, J Basso, A Bengtsson, G Bortoli, A Bourassa, L Brill, M Broughton, BB Buckley, DA Buell, B Burkett, N Bushnell, Z Chen, B Chiaro, R Collins, P Conner, W Courtney, AL Crook, DM Debroy, S Demura, A Dunsworth, D Eppens, C Erickson, L Faoro, E Farhi, R Fatemi, L Flores, E Forati, AG Fowler, W Giang, C Gidney, D Gilboa, M Giustina, AG Dau, JA Gross, S Habegger, MP Harrigan, M Hoffmann, S Hong, T Huang, A Huff, WJ Huggins, LB Ioffe, SV Isakov, J Iveland, E Jeffrey, Z Jiang, C Jones, D Kafri, K Kechedzhi, T Khattar, S Kim, AY Kitaev, PV Klimov, AR Klots, AN Korotkov, F Kostritsa, JM Kreikebaum, D Landhuis, P Laptev, K-M Lau, J Lee, L Laws, W Liu, A Locharla, O Martin, JR McClean, M McEwen, B Meurer Costa, KC Miao, M Mohseni, S Montazeri, A Morvan, E Mount, W Mruczkiewicz, O Naaman, M Neeley, C Neill, M Newman, TE O’Brien, A Opremcak, A Petukhov, R Potter, C Quintana, NC Rubin, N Saei, D Sank, K Sankaragomathi, KJ Satzinger, C Schuster, MJ Shearn, V Shvarts, D Strain, Y Su, M Szalay, G Vidal, B Villalonga, C Vollgraff-Heidweiller, T White, Z Yao, P Yeh, J Yoo, A Zalcman, Y Zhang, N Zhu, H Neven, D Bacon, J Hilton, E Lucero, R Babbush, S Boixo, A Megrant, Y Chen, J Kelly, V Smelyanskiy, DA Abanin, P Roushan · Science 378 (6621), 785-790, 2022

Inherent symmetry of a quantum system may protect its otherwise fragile states. Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the one-dimensional kicked Ising model, which exhibits nonlocal Majorana edge modes (MEMs) with parity symmetry. We find that any multiqubit Pauli operator overlapping with the MEMs exhibits a uniform late-time decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This characteristic allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Furthermore, the MEMs are found to be resilient against certain symmetry-breaking noise owing to a prethermalization mechanism. Our work elucidates the complex interplay between noise and symmetry-protected edge modes in a solid-state environment.

Joshua J Goings, Alec White, Joonho Lee, Christofer S Tautermann, Matthias Degroote, Craig Gidney, Toru Shiozaki, Ryan Babbush, Nicholas C Rubin · Proceedings of the National Academy of Sciences 119 (38), e2203533119, 2022

An accurate assessment of how quantum computers can be used for chemical simulation, especially their potential computational advantages, provides important context on how to deploy these future devices. To perform this assessment reliably, quantum resource estimates must be coupled with classical computations attempting to answer relevant chemical questions and to define the classical algorithms simulation frontier. Herein, we explore the quantum computation and classical computation resources required to assess the electronic structure of cytochrome P450 enzymes (CYPs) and thus define a classical–quantum advantage boundary. This is accomplished by analyzing the convergence of density matrix renormalization group plus n-electron valence state perturbation theory (DMRG+NEVPT2) and coupled-cluster singles doubles with noniterative triples [CCSD(T)] calculations for spin gaps in models of …

Ruslan N Tazhigulov, Shi-Ning Sun, Reza Haghshenas, Huanchen Zhai, Adrian TK Tan, Nicholas C Rubin, Ryan Babbush, Austin J Minnich, Garnet Kin-Lic Chan · PRX Quantum 3 (4), 040318, 2022

Simulating complex molecules and materials is an anticipated application of quantum devices. With the emergence of hardware designed to target strong quantum advantage in artificial tasks, we examine how the same hardware behaves in modeling physical problems of correlated electronic structure. We simulate static and dynamical electronic structure on a superconducting quantum processor derived from Google’s Sycamore architecture for two representative correlated electron problems: the nitrogenase iron-sulfur molecular clusters and α-ruthenium trichloride, a proximate spin-liquid material. To do so, we simplify the electronic structure into low-energy spin models that fit on the device. With extensive error mitigation and assistance from classical recompilation and simulated data, we achieve quantitatively meaningful results deploying about one fifth of the gate resources used in artificial quantum …

Alexis Morvan, TI Andersen, Xiao Mi, Charles Neill, Andre Petukhov, Kostyantyn Kechedzhi, DA Abanin, Alexios Michailidis, R Acharya, F Arute, K Arya, A Asfaw, J Atalaya, JC Bardin, J Basso, A Bengtsson, G Bortoli, A Bourassa, J Bovaird, L Brill, M Broughton, BB Buckley, DA Buell, T Burger, B Burkett, N Bushnell, Z Chen, B Chiaro, R Collins, P Conner, W Courtney, AL Crook, B Curtin, DM Debroy, A Del Toro Barba, S Demura, A Dunsworth, D Eppens, C Erickson, L Faoro, E Farhi, R Fatemi, L Flores Burgos, E Forati, AG Fowler, B Foxen, W Giang, C Gidney, D Gilboa, M Giustina, A Grajales Dau, JA Gross, S Habegger, MC Hamilton, MP Harrigan, SD Harrington, M Hoffmann, S Hong, T Huang, A Huff, WJ Huggins, SV Isakov, J Iveland, E Jeffrey, Z Jiang, C Jones, P Juhas, D Kafri, T Khattar, M Khezri, M Kieferová, S Kim, AY Kitaev, PV Klimov, AR Klots, AN Korotkov, F Kostritsa, JM Kreikebaum, D Landhuis, P Laptev, K-M Lau, L Laws, J Lee, KW Lee, BJ Lester, AT Lill, W Liu, A Locharla, F Malone, O Martin, JR McClean, M McEwen, B Meurer Costa, KC Miao, M Mohseni, S Montazeri, E Mount, W Mruczkiewicz, O Naaman, M Neeley, A Nersisyan, M Newman, A Nguyen, M Nguyen, MY Niu, TE O’Brien, R Olenewa, A Opremcak, R Potter, C Quintana, NC Rubin, N Saei, D Sank, K Sankaragomathi, KJ Satzinger, HF Schurkus, C Schuster, MJ Shearn, A Shorter, V Shvarts, J Skruzny, WC Smith, D Strain, G Sterling, Y Su, M Szalay, A Torres, G Vidal, B Villalonga, C Vollgraff-Heidweiller, T White, C Xing, Z Yao, P Yeh, J Yoo, A Zalcman, Y Zhang, N Zhu, H Neven, D Bacon, J Hilton, E Lucero, R Babbush, S Boixo, A Megrant, J Kelly, Y Chen, V Smelyanskiy, I Aleiner, LB Ioffe · Nature 612 (7939), 240-245, 2022

Systems of correlated particles appear in many fields of modern science and represent some of the most intractable computational problems in nature. The computational challenge in these systems arises when interactions become comparable to other energy scales, which makes the state of each particle depend on all other particles. The lack of general solutions for the three-body problem and acceptable theory for strongly correlated electrons shows that our understanding of correlated systems fades when the particle number or the interaction strength increases. One of the hallmarks of interacting systems is the formation of multiparticle bound states–. Here we develop a high-fidelity parameterizable fSim gate and implement the periodic quantum circuit of the spin-½ XXZ model in a ring of 24 superconducting qubits. We study the propagation of these excitations and observe their bound nature for up to five …

Craig Gidney, Michael Newman, Matt McEwen · Quantum 6, 813, 2022

We improve the planar honeycomb code by describing boundaries that need no additional physical connectivity, and by optimizing the shape of the qubit patch. We then benchmark the code using Monte Carlo sampling to estimate logical error rates and derive metrics including thresholds, lambdas, and teraquop qubit counts. We determine that the planar honeycomb code can create a logical qubit with one-in-a-trillion logical error rates using 7000 physical qubits at a 0.1% gate-level error rate (or 900 physical qubits given native two-qubit parity measurements). Our results cement the honeycomb code as a promising candidate for two-dimensional qubit architectures with sparse connectivity.

William J Huggins, Kianna Wan, Jarrod McClean, Thomas E O’Brien, Nathan Wiebe, Ryan Babbush · Physical Review Letters 129 (24), 240501, 2022

Many quantum algorithms involve the evaluation of expectation values. Optimal strategies for estimating a single expectation value are known, requiring a number of state preparations that scales with the target error ϵ as O (1/ϵ). In this Letter, we address the task of estimating the expectation values of M different observables, each to within additive error ϵ, with the same 1/ϵ dependence. We describe an approach that leverages Gilyén et al.’s quantum gradient estimation algorithm to achieve O (M/ϵ) scaling up to logarithmic factors, regardless of the commutation properties of the M observables. We prove that this scaling is worst-case optimal in the high-precision regime if the state preparation is treated as a black box, even when the operators are mutually commuting. We highlight the flexibility of our approach by presenting several generalizations, including a strategy for accelerating the estimation of a collection …

Thomas E O'Brien, G Anselmetti, Fotios Gkritsis, VE Elfving, Stefano Polla, William J Huggins, Oumarou Oumarou, Kostyantyn Kechedzhi, Dmitry Abanin, Rajeev Acharya, Igor Aleiner, Richard Allen, Trond Ikdahl Andersen, Kyle Anderson, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Dave Bacon, JC Bardin, Andreas Bengtsson, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob Buckley, DA Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Juan Campero, Yu Chen, Zijun Chen, Ben Chiaro, Desmond Chik, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, AL Crook, Ben Curtin, DM Debroy, S Demura, I Drozdov, A Dunsworth, C Erickson, L Faoro, E Farhi, R Fatemi, VS Ferreira, L Flores Burgos, E Forati, AG Fowler, B Foxen, W Giang, C Gidney, D Gilboa, M Giustina, R Gosula, A Grajales Dau, JA Gross, S Habegger, MC Hamilton, M Hansen, MP Harrigan, SD Harrington, P Heu, J Hilton, MR Hoffmann, S Hong, T Huang, A Huff, LB Ioffe, SV Isakov, J Iveland, E Jeffrey, Z Jiang, C Jones, P Juhas, D Kafri, J Kelly, T Khattar, M Khezri, M Kieferová, S Kim, PV Klimov, AR Klots, R Kothari, AN Korotkov, F Kostritsa, JM Kreikebaum, D Landhuis, P Laptev, K Lau, L Laws, J Lee, K Lee, BJ Lester, AT Lill, W Liu, WP Livingston, A Locharla, E Lucero, FD Malone, S Mandra, O Martin, S Martin, JR McClean, T McCourt, M McEwen, A Megrant, X Mi, A Mieszala, KC Miao, M Mohseni, S Montazeri, A Morvan, R Movassagh, W Mruczkiewicz, O Naaman, M Neeley, C Neill, A Nersisyan, H Neven, M Newman, JH Ng, A Nguyen, M Nguyen, MY Niu, S Omonije, A Opremcak, A Petukhov, R Potter, LP Pryadko, C Quintana, C Rocque, P Roushan, N Saei, D Sank, K Sankaragomathi, KJ Satzinger, HF Schurkus, C Schuster, MJ Shearn, Aaron Shorter, Noah Shutty, V Shvarts · arXiv preprint arXiv:2210.10799, 2022

An important measure of the development of quantum computing platforms has been the simulation of increasingly complex physical systems. Prior to fault-tolerant quantum computing, robust error mitigation strategies are necessary to continue this growth. Here, we study physical simulation within the seniority-zero electron pairing subspace, which affords both a computational stepping stone to a fully correlated model, and an opportunity to validate recently introduced ``purification-based'' error-mitigation strategies. We compare the performance of error mitigation based on doubling quantum resources in time (echo verification) or in space (virtual distillation), on up to qubits of a superconducting qubit quantum processor. We observe a reduction of error by one to two orders of magnitude below less sophisticated techniques (e.g. post-selection); the gain from error mitigation is seen to increase with the system size. Employing these error mitigation strategies enables the implementation of the largest variational algorithm for a correlated chemistry system to-date. Extrapolating performance from these results allows us to estimate minimum requirements for a beyond-classical simulation of electronic structure. We find that, despite the impressive gains from purification-based error mitigation, significant hardware improvements will be required for classically intractable variational chemistry simulations.

Ofer Naaman, José Aumentado · PRX Quantum 3 (2), 020201, 2022

We show that a common language can be used to unify the description of parametrically coupled circuits—parametric amplifiers, frequency converters, and parametric nonreciprocal devices—with that of band-pass filter and impedance matching networks. This enables one to readily adapt network synthesis methods from microwave engineering in the design of parametrically coupled devices having prescribed transfer characteristics, eg, gain, bandwidth, return loss, and isolation. We review basic practical aspects of coupled-mode theory and filter synthesis, and then show how to apply both, on an equal footing, to the design of multipole, broadband parametric and nonreciprocal networks. We supplement the discussion with a range of examples and reference designs.

Nicholas C Rubin, Joonho Lee, Ryan Babbush · Journal of Chemical Theory and Computation 18 (3), 1480-1488, 2022

The most efficient known quantum circuits for preparing unitary coupled cluster states and applying Trotter steps of the arbitrary basis electronic structure Hamiltonian involve interleaved sequences of Fermionic Gaussian circuits and Ising interaction-type circuits. These circuits arise from factorizing the two-body operators generating those unitaries as a sum of squared one-body operators that are simulated using product formulas. We introduce a numerical algorithm for performing this factorization that has an iteration complexity no worse than single particle basis transformations of the two-body operators and often results in many times fewer squared one-body operators in the sum of squares, compared to the analytical decompositions. As an application of this numerical procedure, we demonstrate that our protocol can be used to approximate generic unitary coupled cluster operators and prepare the necessary …

Thomas E O'Brien, Michael Streif, Nicholas C Rubin, Raffaele Santagati, Yuan Su, William J Huggins, Joshua J Goings, Nikolaj Moll, Elica Kyoseva, Matthias Degroote, Christofer S Tautermann, Joonho Lee, Dominic W Berry, Nathan Wiebe, Ryan Babbush · Physical Review Research 4 (4), 043210, 2022

While most work on the quantum simulation of chemistry has focused on computing energy surfaces, a similarly important application requiring subtly different algorithms is the computation of energy derivatives. Almost all molecular properties can be expressed an energy derivative, including molecular forces, which are essential for applications such as molecular dynamics simulations. Here, we introduce new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods. Under cost models appropriate for noisy-intermediate scale quantum devices, we demonstrate how low-rank factorization and other tomography schemes can be optimized for energy derivative calculations. We numerically demonstrate that our techniques reduce the number of circuit repetitions required by many orders of magnitude for even modest systems, and that the cost of estimating …

Dominic W Berry, Yuan Su, Casper Gyurik, Robbie King, Joao Basso, Alexander Del Toro Barba, Abhishek Rajput, Nathan Wiebe, Vedran Dunjko, Ryan Babbush · arXiv preprint arXiv:2209.13581, 2022

Lloyd et al. were first to demonstrate the promise of quantum algorithms for computing Betti numbers in persistent homology (a way of characterizing topological features of data sets). Here, we propose, analyze, and optimize an improved quantum algorithm for topological data analysis (TDA) with reduced scaling, including a method for preparing Dicke states based on inequality testing, a more efficient amplitude estimation algorithm using Kaiser windows, and an optimal implementation of eigenvalue projectors based on Chebyshev polynomials. We compile our approach to a fault-tolerant gate set and estimate constant factors in the Toffoli complexity. Relative to the best classical heuristic algorithms, our analysis reveals that super-quadratic quantum speedups are only possible for this problem when targeting a multiplicative error approximation and the Betti number grows asymptotically. Further, we propose a dequantization of the quantum TDA algorithm that shows that having exponentially large dimension and Betti number are necessary, but insufficient conditions, for super-polynomial advantage. We then introduce and analyze specific problem examples for which super-polynomial advantages may be achieved, and argue that quantum circuits with tens of billions of Toffoli gates can solve some seemingly classically intractable instances.

Mostafa Khezri, Alex Opremcak, Zijun Chen, Andreas Bengtsson, Theodore White, Ofer Naaman, Rajeev Acharya, Kyle Anderson, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Joseph C Bardin, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Bob B Buckley, David A Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Juan Campero, Ben Chiaro, Roberto Collins, Alexander L Crook, Ben Curtin, Sean Demura, Andrew Dunsworth, Catherine Erickson, Reza Fatemi, Vinicius S Ferreira, Leslie Flores Burgos, Ebrahim Forati, Brooks Foxen, Gonzalo Garcia, William Giang, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Michael C Hamilton, Sean D Harrington, Paula Heu, Jeremy Hilton, Markus R Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, Justin Iveland, Evan Jeffrey, Julian Kelly, Seon Kim, Paul V Klimov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Lily Laws, Kenny Lee, Brian J Lester, Alexander T Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Steven Martin, Matt McEwen, Anthony Megrant, Xiao Mi, Kevin C Miao, Shirin Montazeri, Alexis Morvan, Matthew Neeley, Charles Neill, Ani Nersisyan, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Rebecca Potter, Chris Quintana, Charles Rocque, Pedram Roushan, Kannan Sankaragomathi, Kevin J Satzinger, Christopher Schuster, Michael J Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, W Clarke Smith, George Sterling, Marco Szalay, Douglas Thor, Alfredo Torres, Bryan WK Woo, Z Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Ningfeng Zhu, Nicholas Zobrist, Daniel Sank, Alexander Korotkov, Yu Chen, Vadim Smelyanskiy · arXiv preprint arXiv:2212.05097, 2022

Superconducting qubits typically use a dispersive readout scheme, where a resonator is coupled to a qubit such that its frequency is qubit-state dependent. Measurement is performed by driving the resonator, where the transmitted resonator field yields information about the resonator frequency and thus the qubit state. Ideally, we could use arbitrarily strong resonator drives to achieve a target signal-to-noise ratio in the shortest possible time. However, experiments have shown that when the average resonator photon number exceeds a certain threshold, the qubit is excited out of its computational subspace, which we refer to as a measurement-induced state transition. These transitions degrade readout fidelity, and constitute leakage which precludes further operation of the qubit in, for example, error correction. Here we study these transitions using a transmon qubit by experimentally measuring their dependence on qubit frequency, average photon number, and qubit state, in the regime where the resonator frequency is lower than the qubit frequency. We observe signatures of resonant transitions between levels in the coupled qubit-resonator system that exhibit noisy behavior when measured repeatedly in time. We provide a semi-classical model of these transitions based on the rotating wave approximation and use it to predict the onset of state transitions in our experiments. Our results suggest the transmon is excited to levels near the top of its cosine potential following a state transition, where the charge dispersion of higher transmon levels explains the observed noisy behavior of state transitions. Moreover, occupation in these higher energy …

Thomas E O’Brien, Lev B Ioffe, Yuan Su, David Fushman, Hartmut Neven, Ryan Babbush, Vadim Smelyanskiy · PRX Quantum 3 (3), 030345, 2022

We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of …

Joseph Bardin · 2022 IEEE International Solid-State Circuits Conference (ISSCC) 65, 422-424, 2022

The Google Quantum AI team's long-term goal is to realize a large-scale fault-tolerant quantum computer. Of the technologies available to implement the quantum processor at the core of such a system, solid-state superconducting circuits based on Josephson junctions (JJs) are among the strongest contenders. Recent demonstrations with this technology include computation beyond the capabilities of today's most powerful supercomputers [1] and execution of quantum error correction (QEC) codes that can detect and correct either bit or phase flip errors [2]. These exciting results were enabled by low-error-rate quantum processors with tens of qubits and the supporting infrastructure to run these devices. However, today's quantum computers still have orders of magnitude fewer qubits than is required for a useful fault-tolerant quantum computer; for this, an estimated million or more qubits will be needed and scaling …

Trond I Andersen, Yuri D Lensky, Kostyantyn Kechedzhi, Ilya Drozdov, Andreas Bengtsson, Sabrina Hong, Alexis Morvan, Xiao Mi, Alex Opremcak, Rajeev Acharya, Richard Allen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B Buckley, David A Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Zijun Chen, Ben Chiaro, Desmond Chik, Charina Chou, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L Crook, Ben Curtin, Dripto M Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Vinicius S Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Jonathan A Gross, Steve Habegger, Michael C Hamilton, Monica Hansen, Matthew P Harrigan, Sean D Harrington, Paula Heu, Jeremy Hilton, Markus R Hoffmann, Trent Huang, Ashley Huff, William J Huggins, Lev B Ioffe, Sergei V Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V Klimov, Andrey R Klots, Alexander N Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D Malone, Orion Martin, Jarrod R McClean, Trevor McCourt, Matt McEwen, Kevin C Miao, Amanda Mieszala, Masoud Mohseni, Shirin Montazeri, Emily Mount, Ramis Movassagh, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E O'Brien, Seun Omonije, Andre Petukhov, Rebecca Potter, Leonid P Pryadko, Chris Quintana, Charles Rocque, Nicholas C Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J Satzinger, Henry F Schurkus, Christopher Schuster, Michael J Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Jindra Skruzny, W Clarke Smith, Rolando Somma, George Sterling, Doug Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White · arXiv preprint arXiv:2210.10255, 2022

Indistinguishability of particles is a fundamental principle of quantum mechanics. For all elementary and quasiparticles observed to date - including fermions, bosons, and Abelian anyons - this principle guarantees that the braiding of identical particles leaves the system unchanged. However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions. Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well developed mathematical description of non-Abelian anyons and numerous theoretical proposals, their experimental observation has remained elusive for decades. Using a superconducting quantum processor, we prepare the ground state of the surface code and manipulate it via unitary operations to form wavefunctions that are described by non-Abelian anyons. By implementing a unitary protocol to move the anyons, we experimentally verify the fusion rules of non-Abelian Ising anyons and braid them to realize their statistics. Building on our technique, we study the prospect of employing the anyons for quantum computation and utilize braiding to create an entangled state of anyons encoding three logical qubits. Our work represents a key step towards topological quantum computing.

Lingling Lao, Alexander Korotkov, Zhang Jiang, Wojciech Mruczkiewicz, Thomas E O'Brien, Dan E Browne · Quantum Science and Technology 7 (2), 025021, 2022

Two-qubit gates are important components of quantum computing. However, unwanted interactions between qubits (so-called parasitic gates) can be particularly problematic and degrade the performance of quantum applications. In this work, we present two software methods to mitigate parasitic two-qubit gate errors. The first approach is built upon the Cartan's KAK decomposition and keeps the original unitary decomposition for the error-free native two-qubit gate. It counteracts a parasitic two-qubit gate by only applying single-qubit rotations and therefore has no two-qubit gate overhead. We show the optimal choice of single-qubit mitigation gates. The second approach applies a numerical optimisation algorithm to re-compile a target unitary into the error-parasitic two-qubit gate plus single-qubit gates. We demonstrate these approaches on the CPhase-parasitic iSWAP-like gates. The KAK-based approach helps …

Eric B Jones, Logan E Hillberry, Matthew T Jones, Mina Fasihi, Pedram Roushan, Zhang Jiang, Alan Ho, Charles Neill, Eric Ostby, Peter Graf, Eliot Kapit, Lincoln D Carr · Nature communications 13 (1), 4483, 2022

Quantum cellular automata (QCA) evolve qubits in a quantum circuit depending only on the states of their neighborhoods and model how rich physical complexity can emerge from a simple set of underlying dynamical rules. The inability of classical computers to simulate large quantum systems hinders the elucidation of quantum cellular automata, but quantum computers offer an ideal simulation platform. Here, we experimentally realize QCA on a digital quantum processor, simulating a one-dimensional Goldilocks rule on chains of up to 23 superconducting qubits. We calculate calibrated and error-mitigated population dynamics and complex network measures, which indicate the formation of small-world mutual information networks. These networks decohere at fixed circuit depth independent of system size, the largest of which corresponding to 1,056 two-qubit gates. Such computations may enable the …

Madelyn Cain, Edward Farhi, Sam Gutmann, Daniel Ranard, Eugene Tang · arXiv preprint arXiv:2207.05089, 2022

The Quantum Approximate Optimization Algorithm (QAOA) is designed to maximize a cost function over bit strings. While the initial state is traditionally a superposition over all strings, it is natural to try expediting the QAOA: first use a classical algorithm to produce some good string, and then run the ordinary QAOA starting in the computational basis state associated with that string. Here we report numerical experiments that this method of initializing the QAOA fails dramatically, exhibiting little to no improvement of the cost function. We investigate criteria for the rare instances in which there is any improvement at all, and we provide a statistical argument for the more typical case of no improvement. The statistical argument holds for any string that locally mimics the thermal ensemble at the appropriate temperature. Our numerical experiments indicate this property holds for typical good strings. We emphasize that our negative results only apply to our simple incarnation of the warm-start QAOA and may not apply to other approaches in the literature. We hope that our theoretical analysis will inform future algorithm design.

Kevin C Miao, Matt McEwen, Juan Atalaya, Dvir Kafri, Leonid P Pryadko, Andreas Bengtsson, Alex Opremcak, Kevin J Satzinger, Zijun Chen, Paul V Klimov, Chris Quintana, Rajeev Acharya, Kyle Anderson, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Joseph C Bardin, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Bob B Buckley, David A Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Juan Campero, Ben Chiaro, Roberto Collins, Paul Conner, Alexander L Crook, Ben Curtin, Dripto M Debroy, Sean Demura, Andrew Dunsworth, Catherine Erickson, Reza Fatemi, Vinicius S Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G Fowler, Brooks Foxen, Gonzalo Garcia, William Giang, Craig Gidney, Marissa Giustina, Raja Gosula, Alejandro Grajales Dau, Jonathan A Gross, Michael C Hamilton, Sean D Harrington, Paula Heu, Jeremy Hilton, Markus R Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Julian Kelly, Seon Kim, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Lily Laws, Kenny Lee, Brian J Lester, Alexander T Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Steven Martin, Anthony Megrant, Xiao Mi, Shirin Montazeri, Alexis Morvan, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Rebecca Potter, Charles Rocque, Pedram Roushan, Kannan Sankaragomathi, Christopher Schuster, Michael J Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Jindra Skruzny, W Clarke Smith, George Sterling, Marco Szalay, Douglas Thor, Alfredo Torres, Theodore White, Bryan WK Woo, Z Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Ningfeng Zhu, Nicholas Zobrist, Hartmut Neven, Vadim Smelyanskiy, Andre Petukhov, Alexander N Korotkov, Daniel Sank, Yu Chen · arXiv preprint arXiv:2211.04728, 2022

Leakage of quantum information out of computational states into higher energy states represents a major challenge in the pursuit of quantum error correction (QEC). In a QEC circuit, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of logical error with scale, challenging the feasibility of QEC as a path towards fault-tolerant quantum computation. Here, we demonstrate the execution of a distance-3 surface code and distance-21 bit-flip code on a Sycamore quantum processor where leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a ten-fold reduction in steady-state leakage population on the data qubits encoding the logical state and an average leakage population of less than throughout the entire device. The leakage removal process itself efficiently returns leakage population back to the computational basis, and adding it to a code circuit prevents leakage from inducing correlated error across cycles, restoring a fundamental assumption of QEC. With this demonstration that leakage can be contained, we resolve a key challenge for practical QEC at scale.

Thomas Schuster, Murphy Niu, Jordan Cotler, Thomas O'Brien, Jarrod R McClean, Masoud Mohseni · arXiv preprint arXiv:2208.02254, 2022

Learning the properties of dynamical quantum systems underlies applications ranging from nuclear magnetic resonance spectroscopy to quantum device characterization. A central challenge in this pursuit is the learning of strongly-interacting systems, where conventional observables decay quickly in time and space, limiting the information that can be learned from their measurement. In this work, we introduce a new class of observables into the context of quantum learning -- the out-of-time-order correlator -- which we show can substantially improve the learnability of strongly-interacting systems by virtue of displaying informative physics at large times and distances. We identify two general scenarios in which out-of-time-order correlators provide a significant advantage for learning tasks in locally-interacting systems: (i) when experimental access to the system is spatially-restricted, for example via a single "probe" degree of freedom, and (ii) when one desires to characterize weak interactions whose strength is much less than the typical interaction strength. We numerically characterize these advantages across a variety of learning problems, and find that they are robust to both read-out error and decoherence. Finally, we introduce a binary classification task that can be accomplished in constant time with out-of-time-order measurements. In a companion paper, we prove that this task is exponentially hard with any adaptive learning protocol that only involves time-ordered operations.

Jordan Cotler, Thomas Schuster, Masoud Mohseni · arXiv preprint arXiv:2208.02256, 2022

We establish that there are properties of quantum many-body dynamics which are efficiently learnable if we are given access to out-of-time-order correlators (OTOCs), but which require exponentially many operations in the system size if we can only measure time-ordered correlators. This implies that any experimental protocol which reconstructs OTOCs solely from time-ordered correlators must be, in certain cases, exponentially inefficient. Our proofs leverage and generalize recent techniques in quantum learning theory. Along the way, we elucidate a general definition of time-ordered versus out-of-time-order experimental measurement protocols, which can be considered as classes of adaptive quantum learning algorithms. Moreover, our results provide a theoretical foundation for novel applications of OTOCs in quantum simulations.

Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, Patrick J Coles · Nature Reviews Physics 3 (9), 625-644, 2021

Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers, owing to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will probably not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational quantum algorithms (VQAs), which use a classical optimizer to train a parameterized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisaged for quantum computers, and they appear to be the best hope for obtaining quantum advantage. Nevertheless, challenges remain, including the trainability, accuracy and efficiency of …

HY Huang, M Broughton, M Mohseni, R Babbush, S Boixo, H Neven, JR McClean · Nature Communications 12, 2631, 2021

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Craig Gidney, Martin Ekerå · Quantum 5, 433, 2021

We significantly reduce the cost of factoring integers and computing discrete logarithms in finite fields on a quantum computer by combining techniques from Shor 1994, Griffiths-Niu 1996, Zalka 2006, Fowler 2012, Ekerå-Håstad 2017, Ekerå 2017, Ekerå 2018, Gidney-Fowler 2019, Gidney 2019. We estimate the approximate cost of our construction using plausible physical assumptions for large-scale superconducting qubit platforms: a planar grid of qubits with nearest-neighbor connectivity, a characteristic physical gate error rate of , a surface code cycle time of 1 microsecond, and a reaction time of 10 microseconds. We account for factors that are normally ignored such as noise, the need to make repeated attempts, and the spacetime layout of the computation. When factoring 2048 bit RSA integers, our construction's spacetime volume is a hundredfold less than comparable estimates from earlier works (Van Meter et al. 2009, Jones et al. 2010, Fowler et al. 2012, Gheorghiu et al. 2019). In the abstract circuit model (which ignores overheads from distillation, routing, and error correction) our construction uses logical qubits, Toffolis, and measurement depth to factor -bit RSA integers. We quantify the cryptographic implications of our work, both for RSA and for schemes based on the DLP in finite fields.

Matthew P Harrigan, Kevin J Sung, Matthew Neeley, Kevin J Satzinger, Frank Arute, Kunal Arya, Juan Atalaya, Joseph C Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B Buckley, David A Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Daniel Eppens, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Alan Ho, Sabrina Hong, Trent Huang, LB Ioffe, Sergei V Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V Klimov, Alexander N Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Martin Leib, Orion Martin, John M Martinis, Jarrod R McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Charles Neill, Florian Neukart, Murphy Yuezhen Niu, Thomas E O’Brien, Bryan O’Gorman, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C Rubin, Daniel Sank, Andrea Skolik, Vadim Smelyanskiy, Doug Strain, Michael Streif, Marco Szalay, Amit Vainsencher, Theodore White, Z Jamie Yao, Ping Yeh, Adam Zalcman, Leo Zhou, Hartmut Neven, Dave Bacon, Erik Lucero, Edward Farhi, Ryan Babbush · Nature Physics 17 (3), 332-336, 2021

Faster algorithms for combinatorial optimization could prove transformative for diverse areas such as logistics, finance and machine learning. Accordingly, the possibility of quantum enhanced optimization has driven much interest in quantum technologies. Here we demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the planar connectivity graph native to our hardware; however, we also apply the QAOA to the Sherrington–Kirkpatrick model and MaxCut, non-native problems that require extensive compilation to implement. For hardware-native problems, which are classically efficient to solve on average, we obtain an approximation ratio that is independent of problem size and observe that …

Ehud Altman, Kenneth R Brown, Giuseppe Carleo, Lincoln D Carr, Eugene Demler, Cheng Chin, Brian DeMarco, Sophia E Economou, Mark A Eriksson, Kai-Mei C Fu, Markus Greiner, Kaden RA Hazzard, Randall G Hulet, Alicia J Kollár, Benjamin L Lev, Mikhail D Lukin, Ruichao Ma, Xiao Mi, Shashank Misra, Christopher Monroe, Kater Murch, Zaira Nazario, Kang-Kuen Ni, Andrew C Potter, Pedram Roushan, Mark Saffman, Monika Schleier-Smith, Irfan Siddiqi, Raymond Simmonds, Meenakshi Singh, IB Spielman, Kristan Temme, David S Weiss, Jelena Vučković, Vladan Vuletić, Jun Ye, Martin Zwierlein · PRX Quantum 2 (1), 017003, 2021

Quantum simulators are a promising technology on the spectrum of quantum devices from specialized quantum experiments to universal quantum computers. These quantum devices utilize entanglement and many-particle behavior to explore and solve hard scientific, engineering, and computational problems. Rapid development over the last two decades has produced more than 300 quantum simulators in operation worldwide using a wide variety of experimental platforms. Recent advances in several physical architectures promise a golden age of quantum simulators ranging from highly optimized special purpose simulators to flexible programmable devices. These developments have enabled a convergence of ideas drawn from fundamental physics, computer science, and device engineering. They have strong potential to address problems of societal importance, ranging from understanding vital chemical …

KJ Satzinger, Y-J Liu, A Smith, C Knapp, M Newman, C Jones, Z Chen, C Quintana, X Mi, A Dunsworth, C Gidney, I Aleiner, F Arute, K Arya, J Atalaya, R Babbush, JC Bardin, R Barends, J Basso, A Bengtsson, A Bilmes, M Broughton, BB Buckley, DA Buell, B Burkett, N Bushnell, B Chiaro, R Collins, W Courtney, S Demura, AR Derk, D Eppens, C Erickson, L Faoro, E Farhi, AG Fowler, B Foxen, M Giustina, A Greene, JA Gross, MP Harrigan, SD Harrington, J Hilton, S Hong, T Huang, WJ Huggins, LB Ioffe, SV Isakov, E Jeffrey, Z Jiang, D Kafri, K Kechedzhi, T Khattar, S Kim, PV Klimov, AN Korotkov, F Kostritsa, D Landhuis, P Laptev, A Locharla, E Lucero, O Martin, JR McClean, M McEwen, KC Miao, M Mohseni, S Montazeri, W Mruczkiewicz, J Mutus, O Naaman, M Neeley, C Neill, MY Niu, TE O’Brien, A Opremcak, B Pató, A Petukhov, NC Rubin, D Sank, V Shvarts, D Strain, M Szalay, B Villalonga, TC White, Z Yao, P Yeh, J Yoo, A Zalcman, H Neven, S Boixo, A Megrant, Y Chen, J Kelly, V Smelyanskiy, A Kitaev, M Knap, F Pollmann, P Roushan · Science 374 (6572), 1237-1241, 2021

The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error–correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of –ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.

Nature 595 (7867), 383-387, 2021

Realizing the potential of quantum computing requires sufficiently low logical error rates. Many applications call for error rates as low as 10−15 (refs. –), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. –). Quantum error correction– promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the …

Andrea Skolik, Jarrod R McClean, Masoud Mohseni, Patrick van der Smagt, Martin Leib · Quantum Machine Intelligence 3, 1-11, 2021

With the increased focus on quantum circuit learning for near-term applications on quantum devices, in conjunction with unique challenges presented by cost function landscapes of parametrized quantum circuits, strategies for effective training are becoming increasingly important. In order to ameliorate some of these challenges, we investigate a layerwise learning strategy for parametrized quantum circuits. The circuit depth is incrementally grown during optimization, and only subsets of parameters are updated in each training step. We show that when considering sampling noise, this strategy can help avoid the problem of barren plateaus of the error surface due to the low depth of circuits, low number of parameters trained in one step, and larger magnitude of gradients compared to training the full circuit. These properties make our algorithm preferable for execution on noisy intermediate-scale quantum …

Xiao Mi, Pedram Roushan, Chris Quintana, Salvatore Mandra, Jeffrey Marshall, Charles Neill, Frank Arute, Kunal Arya, Juan Atalaya, Ryan Babbush, Joseph C Bardin, Rami Barends, Joao Basso, Andreas Bengtsson, Sergio Boixo, Alexandre Bourassa, Michael Broughton, Bob B Buckley, David A Buell, Brian Burkett, Nicholas Bushnell, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Alan R Derk, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Edward Farhi, Austin G Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Jonathan A Gross, Matthew P Harrigan, Sean D Harrington, Jeremy Hilton, Alan Ho, Sabrina Hong, Trent Huang, William J Huggins, LB Ioffe, Sergei V Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Julian Kelly, Seon Kim, Alexei Kitaev, Paul V Klimov, Alexander N Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Erik Lucero, Orion Martin, Jarrod R McClean, Trevor McCourt, Matt McEwen, Anthony Megrant, Kevin C Miao, Masoud Mohseni, Shirin Montazeri, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Michael Newman, Murphy Yuezhen Niu, Thomas E O’Brien, Alex Opremcak, Eric Ostby, Balint Pato, Andre Petukhov, Nicholas Redd, Nicholas C Rubin, Daniel Sank, Kevin J Satzinger, Vladimir Shvarts, Doug Strain, Marco Szalay, Matthew D Trevithick, Benjamin Villalonga, Theodore White, Z Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, Igor Aleiner, Kostyantyn Kechedzhi, Vadim Smelyanskiy, Yu Chen · Science 374 (6574), 1479-1483, 2021

Interactions in quantum systems can spread initially localized quantum information into the exponentially many degrees of freedom of the entire system. Understanding this process, known as quantum scrambling, is key to resolving several open questions in physics. Here, by measuring the time-dependent evolution and fluctuation of out-of-time-order correlators, we experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We engineer quantum circuits that distinguish operator spreading and operator entanglement and experimentally observe their respective signatures. We show that whereas operator spreading is captured by an efficient classical model, operator entanglement in idealized circuits requires exponentially scaled computational resources to simulate. These results open the path to studying complex and practically relevant physical observables with near-term …

William J Huggins, Jarrod R McClean, Nicholas C Rubin, Zhang Jiang, Nathan Wiebe, K Birgitta Whaley, Ryan Babbush · npj Quantum Information 7 (1), 23, 2021

Variational algorithms are a promising paradigm for utilizing near-term quantum devices for modeling electronic states of molecular systems. However, previous bounds on the measurement time required have suggested that the application of these techniques to larger molecules might be infeasible. We present a measurement strategy based on a low-rank factorization of the two-electron integral tensor. Our approach provides a cubic reduction in term groupings over prior state-of-the-art and enables measurement times three orders of magnitude smaller than those suggested by commonly referenced bounds for the largest systems we consider. Although our technique requires execution of a linear-depth circuit prior to measurement, this is compensated for by eliminating challenges associated with sampling nonlocal Jordan–Wigner transformed operators in the presence of measurement error, while enabling a …

William J Huggins, Sam McArdle, Thomas E O’Brien, Joonho Lee, Nicholas C Rubin, Sergio Boixo, K Birgitta Whaley, Ryan Babbush, Jarrod R McClean · Physical Review X 11 (4), 041036, 2021

Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits. Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then, it will be necessary to use alternative strategies to mitigate the impact of errors. We propose a near-term friendly strategy to mitigate errors by entangling and measuring M copies of a noisy state ρ. This enables us to estimate expectation values with respect to a state with dramatically reduced error ρ M/Tr (ρ M) without explicitly preparing it, hence the name “virtual distillation.” As M increases, this state approaches the closest pure state to ρ exponentially quickly. We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behavior of this pure state (corresponding to the dominant …

Joonho Lee, Dominic W Berry, Craig Gidney, William J Huggins, Jarrod R McClean, Nathan Wiebe, Ryan Babbush · PRX Quantum 2 (3), 030305, 2021

We describe quantum circuits with only O~(N) Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of N arbitrary (eg, molecular) orbitals. With O (λ/ϵ) repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where λ is the 1-norm of Hamiltonian coefficients and ϵ is the target precision. This is the lowest complexity shown for quantum computations of chemistry within an arbitrary basis. Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can work only with orthogonal-basis functions diagonalizing the Coloumb operator (eg, the plane-wave dual basis). Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a nonorthogonal basis defined by the THC …

Andrew D King, Jack Raymond, Trevor Lanting, Sergei V Isakov, Masoud Mohseni, Gabriel Poulin-Lamarre, Sara Ejtemaee, William Bernoudy, Isil Ozfidan, Anatoly Yu Smirnov, Mauricio Reis, Fabio Altomare, Michael Babcock, Catia Baron, Andrew J Berkley, Kelly Boothby, Paul I Bunyk, Holly Christiani, Colin Enderud, Bram Evert, Richard Harris, Emile Hoskinson, Shuiyuan Huang, Kais Jooya, Ali Khodabandelou, Nicolas Ladizinsky, Ryan Li, P Aaron Lott, Allison JR MacDonald, Danica Marsden, Gaelen Marsden, Teresa Medina, Reza Molavi, Richard Neufeld, Mana Norouzpour, Travis Oh, Igor Pavlov, Ilya Perminov, Thomas Prescott, Chris Rich, Yuki Sato, Benjamin Sheldan, George Sterling, Loren J Swenson, Nicholas Tsai, Mark H Volkmann, Jed D Whittaker, Warren Wilkinson, Jason Yao, Hartmut Neven, Jeremy P Hilton, Eric Ladizinsky, Mark W Johnson, Mohammad H Amin · Nature communications 12 (1), 1113, 2021

The promise of quantum computing lies in harnessing programmable quantum devices for practical applications such as efficient simulation of quantum materials and condensed matter systems. One important task is the simulation of geometrically frustrated magnets in which topological phenomena can emerge from competition between quantum and thermal fluctuations. Here we report on experimental observations of equilibration in such simulations, measured on up to 1440 qubits with microsecond resolution. By initializing the system in a state with topological obstruction, we observe quantum annealing (QA) equilibration timescales in excess of one microsecond. Measurements indicate a dynamical advantage in the quantum simulation compared with spatially local update dynamics of path-integral Monte Carlo (PIMC). The advantage increases with both system size and inverse temperature, exceeding a …

Ryan Babbush, Jarrod R McClean, Michael Newman, Craig Gidney, Sergio Boixo, Hartmut Neven · PRX Quantum 2 (1), 010103, 2021

In this perspective we discuss conditions under which it would be possible for a modest fault-tolerant quantum computer to realize a runtime advantage by executing a quantum algorithm with only a small polynomial speedup over the best classical alternative. The challenge is that the computation must finish within a reasonable amount of time while being difficult enough that the small quantum scaling advantage would compensate for the large constant factor overheads associated with error correction. We compute several examples of such runtimes using state-of-the-art surface code constructions under a variety of assumptions. We conclude that quadratic speedups will not enable quantum advantage on early generations of such fault-tolerant devices unless there is a significant improvement in how we realize quantum error correction. While this conclusion persists even if we were to increase the rate of logical …

Craig Gidney · Quantum 5, 497, 2021

This paper presents “Stim", a fast simulator for quantum stabilizer circuits. The paper explains how Stim works and compares it to existing tools. With no foreknowledge, Stim can analyze a distance 100 surface code circuit (20 thousand qubits, 8 million gates, 1 million measurements) in 15 seconds and then begin sampling full circuit shots at a rate of 1 kHz. Stim uses a stabilizer tableau representation, similar to Aaronson and Gottesman's CHP simulator, but with three main improvements. First, Stim improves the asymptotic complexity of deterministic measurement from quadratic to linear by tracking the $ inverse $ of the circuit's stabilizer tableau. Second, Stim improves the constant factors of the algorithm by using a cache-friendly data layout and 256 bit wide SIMD instructions. Third, Stim only uses expensive stabilizer tableau simulation to create an initial reference sample. Further samples are collected in bulk by using that sample as a reference for batches of Pauli frames propagating through the circuit.

Mario Motta, Erika Ye, Jarrod R McClean, Zhendong Li, Austin J Minnich, Ryan Babbush, Garnet Kin-Lic Chan · npj Quantum Information 7 (1), 83, 2021

The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for N molecular orbitals, the O (N 4) gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with O (N 3) gate complexity in small simulations, which reduces to O (N 2) gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with O (N 3) gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits …

Alicia B Magann, Christian Arenz, Matthew D Grace, Tak-San Ho, Robert L Kosut, Jarrod R McClean, Herschel A Rabitz, Mohan Sarovar · PRX Quantum 2 (1), 010101, 2021

The last decade has witnessed remarkable progress in the development of quantum technologies. Although fault-tolerant devices likely remain years away, the noisy intermediate-scale quantum devices of today may be leveraged for other purposes. Leading candidates are variational quantum algorithms (VQAs), which have been developed for applications including chemistry, optimization, and machine learning, but whose implementations on quantum devices have yet to demonstrate improvements over classical capabilities. In this Perspective, we propose a variety of ways that the performance of VQAs could be informed by quantum optimal control theory. A major theme throughout is the need for sufficient control resources in VQA implementations; we discuss different ways this need can manifest, outline a variety of open questions, and look to the future.

Andrew Zhao, Nicholas C Rubin, Akimasa Miyake · Physical Review Letters 127 (11), 110504, 2021

We propose a tomographic protocol for estimating any k-body reduced density matrix (k-RDM) of an n-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all k-RDM elements to additive precision ϵ requires on the order of (n k) k 3/2 log (n)/ϵ 2 repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for k≥ 2, as compared to prior deterministic strategies …

Matt McEwen, Dvir Kafri, Z Chen, Juan Atalaya, KJ Satzinger, Chris Quintana, Paul Victor Klimov, Daniel Sank, C Gidney, AG Fowler, F Arute, Kunal Arya, B Buckley, Brian Burkett, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, Sean Demura, Andrew Dunsworth, Catherine Erickson, B Foxen, Marissa Giustina, Trent Huang, Sabrina Hong, Evan Jeffrey, Seon Kim, Kostyantyn Kechedzhi, Fedor Kostritsa, Pavel Laptev, Anthony Megrant, Xiao Mi, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, M Niu, Alexandru Paler, Nick Redd, Pedram Roushan, TC White, Jamie Yao, Ping Yeh, A Zalcman, Yu Chen, VN Smelyanskiy, John M Martinis, Hartmut Neven, J Kelly, AN Korotkov, Andre Gregory Petukhov, Rami Barends · Nature communications 12 (1), 1761, 2021

Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key …

C Neill, T McCourt, X Mi, Z Jiang, MY Niu, W Mruczkiewicz, I Aleiner, F Arute, K Arya, J Atalaya, R Babbush, JC Bardin, R Barends, A Bengtsson, A Bourassa, M Broughton, BB Buckley, DA Buell, B Burkett, N Bushnell, J Campero, Z Chen, B Chiaro, R Collins, W Courtney, S Demura, AR Derk, A Dunsworth, D Eppens, C Erickson, E Farhi, AG Fowler, B Foxen, C Gidney, M Giustina, JA Gross, MP Harrigan, SD Harrington, J Hilton, A Ho, S Hong, T Huang, WJ Huggins, SV Isakov, M Jacob-Mitos, E Jeffrey, C Jones, D Kafri, K Kechedzhi, J Kelly, S Kim, PV Klimov, AN Korotkov, F Kostritsa, D Landhuis, P Laptev, E Lucero, O Martin, JR McClean, M McEwen, A Megrant, KC Miao, M Mohseni, J Mutus, O Naaman, M Neeley, M Newman, TE O’Brien, A Opremcak, E Ostby, B Pató, A Petukhov, C Quintana, N Redd, NC Rubin, D Sank, KJ Satzinger, V Shvarts, D Strain, M Szalay, MD Trevithick, B Villalonga, TC White, Z Yao, P Yeh, A Zalcman, H Neven, S Boixo, LB Ioffe, P Roushan, Y Chen, V Smelyanskiy · Nature 594 (7864), 508-512, 2021

A promising approach to study condensed-matter systems is to simulate them on an engineered quantum platform–. However, the accuracy needed to outperform classical methods has not been achieved so far. Here, using 18 superconducting qubits, we provide an experimental blueprint for an accurate condensed-matter simulator and demonstrate how to investigate fundamental electronic properties. We benchmark the underlying method by reconstructing the single-particle band structure of a one-dimensional wire. We demonstrate nearly complete mitigation of decoherence and readout errors, and measure the energy eigenvalues of this wire with an error of approximately 0.01 rad, whereas typical energy scales are of the order of 1 rad. Insight into the fidelity of this algorithm is gained by highlighting the robust properties of a Fourier transform, including the ability to resolve eigenenergies with a statistical …

Yuan Su, Dominic W Berry, Nathan Wiebe, Nicholas Rubin, Ryan Babbush · PRX Quantum 2 (4), 040332, 2021

Quantum simulations of chemistry in first quantization offer some important advantages over approaches in second quantization including faster convergence to the continuum limit and the opportunity for practical simulations outside of the Born-Oppenheimer approximation. However, since all prior work on quantum simulation of chemistry in first quantization has been limited to asymptotic analysis, it has been impossible to directly compare the resources required for these approaches to those required for the more commonly studied algorithms in second quantization. Here, we compile, optimize, and analyze the finite resources required to implement two first quantized quantum algorithms for chemistry from Babbush et al.[Npj Quantum Inf. 5, 92 (2019)] that realize block encodings for the qubitization and interaction-picture frameworks of Low et al.[Quantum 3, 163 (2019), arXiv: 1805.00675 (2018)]. The two …

Thomas E O’Brien, Stefano Polla, Nicholas C Rubin, William J Huggins, Sam McArdle, Sergio Boixo, Jarrod R McClean, Ryan Babbush · PRX Quantum 2 (2), 020317, 2021

The accumulation of noise in quantum computers is the dominant issue stymieing the push of quantum algorithms beyond their classical counterparts. We do not expect to be able to afford the overhead required for quantum error correction in the next decade, so in the meantime we must rely on low-cost, unscalable error mitigation techniques to bring quantum computing to its full potential. In this paper we present a new error mitigation technique based on quantum phase estimation that can also reduce errors in expectation value estimation (eg, for variational algorithms). The general idea is to apply phase estimation while effectively postselecting for the system register to be in the starting state, which allows us to catch and discard errors that knock us away from there. We refer to this technique as “verified phase estimation”(VPE) and show that it can be adapted to function without the use of control qubits in order to …

Hongxiang Chen, Leonard Wossnig, Simone Severini, Hartmut Neven, Masoud Mohseni · Quantum Machine Intelligence 3, 1-11, 2021

Recent results have demonstrated the successful applications of quantum-classical hybrid methods to train quantum circuits for a variety of machine learning tasks. A natural question to ask is consequentially whether we can also train such quantum circuits to discriminate quantum data, i.e., perform classification on data stored in form of quantum states. Although quantum mechanics fundamentally forbids deterministic discrimination of non-orthogonal states, we show in this work that it is possible to train a quantum circuit to discriminate such data with a trade-off between minimizing error rates and inconclusiveness rates of the classification tasks. Our approach achieves at the same time a performance which is close to the theoretically optimal values and a generalization ability to previously unseen quantum data. This generalization power hence distinguishes our work from previous circuit optimization …

Evan Peters, João Caldeira, Alan Ho, Stefan Leichenauer, Masoud Mohseni, Hartmut Neven, Panagiotis Spentzouris, Doug Strain, Gabriel N Perdue · npj Quantum Information 7 (1), 161, 2021

Quantum kernel methods show promise for accelerating data analysis by efficiently learning relationships between input data points that have been encoded into an exponentially large Hilbert space. While this technique has been used successfully in small-scale experiments on synthetic datasets, the practical challenges of scaling to large circuits on noisy hardware have not been thoroughly addressed. Here, we present our findings from experimentally implementing a quantum kernel classifier on real high-dimensional data taken from the domain of cosmology using Google’s universal quantum processor, Sycamore. We construct a circuit ansatz that preserves kernel magnitudes that typically otherwise vanish due to an exponentially growing Hilbert space, and implement error mitigation specific to the task of computing quantum kernels on near-term hardware. Our experiment utilizes 17 qubits to classify …

Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, Leo Zhou · arXiv preprint arXiv:2110.14206, 2021

The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth . We apply the QAOA to MaxCut on large-girth -regular graphs. We give an iterative formula to evaluate performance for any at any depth . Looking at random -regular graphs, at optimal parameters and as goes to infinity, we find that the QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these -regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max--XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as . This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to . Encouraged by our findings, we make the optimistic conjecture that the QAOA, as goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.

Craig Gidney, Michael Newman, Austin Fowler, Michael Broughton · Quantum 5, 605, 2021

Recently, Hastings & Haah introduced a quantum memory defined on the honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity checks using only two-local measurements. The sparse connectivity and two-local measurements are desirable features for certain hardware, while the weight-six parity checks enable robust performance in the circuit model.

Jarrod R McClean, Matthew P Harrigan, Masoud Mohseni, Nicholas C Rubin, Zhang Jiang, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, Hartmut Neven · PRX Quantum 2 (3), 030312, 2021

One of the major application areas of interest for both near-term and fault-tolerant quantum computers is the optimization of classical objective functions. In this work, we develop intuitive constructions for a large class of these algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. This is connected to effects from wavefunction confinement, phase randomization, and shadow defects lurking in the objective far away from the ideal solution. As an example, we explore …

Benjamin Villalonga, Murphy Yuezhen Niu, Li Li, Hartmut Neven, John C Platt, Vadim N Smelyanskiy, Sergio Boixo · arXiv preprint arXiv:2109.11525, 2021

Two recent landmark experiments have performed Gaussian boson sampling (GBS) with a non-programmable linear interferometer and threshold detectors on up to 144 output modes (see Refs.~\onlinecite{zhong_quantum_2020,zhong2021phase}). Here we give classical sampling algorithms with better total variation distance and Kullback-Leibler divergence than these experiments and a computational cost quadratic in the number of modes. Our method samples from a distribution that approximates the single-mode and two-mode ideal marginals of the given Gaussian boson sampler, which are calculated efficiently. One implementation sets the parameters of a Boltzmann machine from the calculated marginals using a mean field solution. This is a 2nd order approximation, with the uniform and thermal approximations corresponding to the 0th and 1st order, respectively. The th order approximation reproduces Ursell functions (also known as connected correlations) up to order with a cost exponential in and high precision, while the experiment exhibits higher order Ursell functions with lower precision. This methodology, like other polynomial approximations introduced previously, does not apply to random circuit sampling because the th order approximation would simply result in the uniform distribution, in contrast to GBS.

Jordan Cotler, Hsin-Yuan Huang, Jarrod R McClean · arXiv preprint arXiv:2112.00811, 2021

It has been shown that the apparent advantage of some quantum machine learning algorithms may be efficiently replicated using classical algorithms with suitable data access -- a process known as dequantization. Existing works on dequantization compare quantum algorithms which take copies of an n-qubit quantum state as input to classical algorithms which have sample and query (SQ) access to the vector . In this note, we prove that classical algorithms with SQ access can accomplish some learning tasks exponentially faster than quantum algorithms with quantum state inputs. Because classical algorithms are a subset of quantum algorithms, this demonstrates that SQ access can sometimes be significantly more powerful than quantum state inputs. Our findings suggest that the absence of exponential quantum advantage in some learning tasks may be due to SQ access being too powerful relative to quantum state inputs. If we compare quantum algorithms with quantum state inputs to classical algorithms with access to measurement data on quantum states, the landscape of quantum advantage can be dramatically different. We remark that when the quantum states are constructed from exponential-size classical data, comparing SQ access and quantum state inputs is appropriate since both require exponential time to prepare.

Haozhi Wang, Suren Singh, CRH McRae, Joseph C Bardin, SX Lin, N Messaoudi, AR Castelli, YJ Rosen, ET Holland, DP Pappas, JY Mutus · Quantum Science and Technology 6 (3), 035015, 2021

Superconducting circuit testing and materials loss characterization requires robust and reliable methods for the extraction of internal and coupling quality factors of microwave resonators. A common method, imposed by limitations on the device design or experimental configuration, is the single-port reflection geometry, ie reflection-mode. However, impedance mismatches in cryogenic systems must be accounted for through calibration of the measurement chain while it is at low temperatures. In this paper, we demonstrate a data-based, single-port calibration using commercial microwave standards and a vector network analyzer with samples at millikelvin temperature in a dilution refrigerator, making this method useful for measurements of quantum phenomena. Finally, we cross reference our data-based, single-port calibration and reflection measurement with over-coupled 2D-and 3D-resonators against well …

Jarrod R McClean, Nicholas C Rubin, Joonho Lee, Matthew P Harrigan, Thomas E O’Brien, Ryan Babbush, William J Huggins, Hsin-Yuan Huang · The Journal of Chemical Physics 155 (15), 2021

With the rapid development of quantum technology, one of the leading applications that has been identified is the simulation of chemistry. Interestingly, even before full scale quantum computers are available, quantum computer science has exhibited a remarkable string of results that directly impact what is possible in a chemical simulation with any computer. Some of these results even impact our understanding of chemistry in the real world. In this Perspective, we take the position that direct chemical simulation is best understood as a digital experiment. While on the one hand, this clarifies the power of quantum computers to extend our reach, it also shows us the limitations of taking such an approach too directly. Leveraging results that quantum computers cannot outpace the physical world, we build to the controversial stance that some chemical problems are best viewed as problems for which no algorithm can …

Nicholas C Rubin, Klaas Gunst, Alec White, Leon Freitag, Kyle Throssell, Garnet Kin-Lic Chan, Ryan Babbush, Toru Shiozaki · Quantum 5, 568, 2021

The fermionic quantum emulator (FQE) is a collection of protocols for emulating quantum dynamics of fermions efficiently taking advantage of common symmetries present in chemical, materials, and condensed-matter systems. The library is fully integrated with the OpenFermion software package and serves as the simulation backend. The FQE reduces memory footprint by exploiting number and spin symmetry along with custom evolution routines for sparse and dense Hamiltonians, allowing us to study significantly larger quantum circuits at modest computational cost when compared against qubit state vector simulators. This release paper outlines the technical details of the simulation methods and key advantages.

Joseph C Bardin · IEEE Solid-State Circuits Magazine 13 (2), 22-35, 2021

Cryogenically cooled low-noise amplifiers (LNAs) have had a profound impact on experimental science. For instance, these amplifiers allow us to communicate with distant spacecraft, probe the history and composition of the universe through radio astronomy, study basic phenomena through low-temperature physics research, and read out the state of quantum systems as required for quantum computing. These devices-which can achieve noise performance within an order of magnitude of the fundamental limits imposed by quantum mechanics-find use from low frequencies through several hundreds of gigahertz. Without cryogenic LNAs, whole branches of experimental science simply could not exist.

Alexander Zlokapa, Hartmut Neven, Seth Lloyd · arXiv preprint arXiv:2107.09200, 2021

Given the success of deep learning in classical machine learning, quantum algorithms for traditional neural network architectures may provide one of the most promising settings for quantum machine learning. Considering a fully-connected feedforward neural network, we show that conditions amenable to classical trainability via gradient descent coincide with those necessary for efficiently solving quantum linear systems. We propose a quantum algorithm to approximately train a wide and deep neural network up to error for a training set of size by performing sparse matrix inversion in time. To achieve an end-to-end exponential speedup over gradient descent, the data distribution must permit efficient state preparation and readout. We numerically demonstrate that the MNIST image dataset satisfies such conditions; moreover, the quantum algorithm matches the accuracy of the fully-connected network. Beyond the proven architecture, we provide empirical evidence for training of a convolutional neural network with pooling.

Dominik Hangleiter, Ingo Roth, Jens Eisert, Pedram Roushan · arXiv preprint arXiv:2108.08319, 2021

The required precision to perform quantum simulations beyond the capabilities of classical computers imposes major experimental and theoretical challenges. Here, we develop a characterization technique to benchmark the implementation precision of a specific quantum simulation task. We infer all parameters of the bosonic Hamiltonian that governs the dynamics of excitations in a two-dimensional grid of nearest-neighbour coupled superconducting qubits. We devise a robust algorithm for identification of Hamiltonian parameters from measured times series of the expectation values of single-mode canonical coordinates. Using super-resolution and denoising methods, we first extract eigenfrequencies of the governing Hamiltonian from the complex time domain measurement; next, we recover the eigenvectors of the Hamiltonian via constrained manifold optimization over the orthogonal group. For five and six coupled qubits, we identify Hamiltonian parameters with sub-MHz precision and construct a spatial implementation error map for a grid of 27 qubits. Our approach enables us to distinguish and quantify the effects of state preparation and measurement errors and show that they are the dominant sources of errors in the implementation. Our results quantify the implementation accuracy of analog dynamics and introduce a diagnostic toolkit for understanding, calibrating, and improving analog quantum processors.

Poulami Das, Aditya Locharla, Cody Jones · arXiv preprint arXiv:2108.06569, 2021

The error rates of quantum devices are orders of magnitude higher than what is needed to run most quantum applications. To close this gap, Quantum Error Correction (QEC) encodes logical qubits and distributes information using several physical qubits. By periodically executing a syndrome extraction circuit on the logical qubits, information about errors (called syndrome) is extracted while running programs. A decoder uses these syndromes to identify and correct errors in real time, which is required to use feedback implemented in quantum algorithms. Unfortunately, software decoders are slow and hardware decoders are fast but less accurate. Thus, almost all QEC studies so far have relied on offline decoding. To enable real-time decoding in near-term QEC, we propose LILLIPUT-- a Lightweight Low Latency Look-Up Table decoder. LILLIPUT consists of two parts-- First, it translates syndromes into error detection events that index into a Look-Up Table (LUT) whose entry provides the error information in real-time. Second, it programs the LUTs with error assignments for all possible error events by running a software decoder offline. LILLIPUT tolerates an error on any operation in the quantum hardware, including gates and measurement, and the number of tolerated errors grows with the size of the code. It needs <7% logic on off-the-shelf FPGAs that allows it to be easily integrated alongside the control and readout circuits in existing systems. LILLIPUT incurs a latency of few nanoseconds and enables real-time decoding. We also propose Compressed LUTs (CLUTs) to reduce the memory needed by LILLIPUT. By exploiting the fact that not all …

Masoud Mohseni, Daniel Eppens, Johan Strumpfer, Raffaele Marino, Vasil Denchev, Alan K Ho, Sergei V Isakov, Sergio Boixo, Federico Ricci-Tersenghi, Hartmut Neven · arXiv preprint arXiv:2111.13628, 2021

Optimizing highly complex cost/energy functions over discrete variables is at the heart of many open problems across different scientific disciplines and industries. A major obstacle is the emergence of many-body effects among certain subsets of variables in hard instances leading to critical slowing down or collective freezing for known stochastic local search strategies. An exponential computational effort is generally required to unfreeze such variables and explore other unseen regions of the configuration space. Here, we introduce a quantum-inspired family of nonlocal Nonequilibrium Monte Carlo (NMC) algorithms by developing an adaptive gradient-free strategy that can efficiently learn key instance-wise geometrical features of the cost function. That information is employed on-the-fly to construct spatially inhomogeneous thermal fluctuations for collectively unfreezing variables at various length scales, circumventing costly exploration versus exploitation trade-offs. We apply our algorithm to two of the most challenging combinatorial optimization problems: random k-satisfiability (k-SAT) near the computational phase transitions and Quadratic Assignment Problems (QAP). We observe significant speedup and robustness over both specialized deterministic solvers and generic stochastic solvers. In particular, for 90% of random 4-SAT instances we find solutions that are inaccessible for the best specialized deterministic algorithm known as Survey Propagation (SP) with an order of magnitude improvement in the quality of solutions for the hardest 10% instances. We also demonstrate two orders of magnitude improvement in time-to-solution over the …

Zhenjie Zou, Mohsen Hosseini, Randy Kwende, Sanjay Raman, Joseph C Bardin · 2021 IEEE MTT-S International Microwave Symposium (IMS), 653-656, 2021

A 3–6 GHz reconfigurable SiGe cryogenic low-noise amplifier has been designed, fabricated, and tested. The integrated circuit features a broadband input-stage followed by a pair of buffered reconfigurable second-order systems. When characterized at a physical temperature of 15K and configured for a broadband response (3–6 GHz), we find that it provides in excess of 35 dB of gain while achieving an average noise temperature of 4.3K from 3–6 GHz and dissipating 1.8mW. By changing the states of the digitally controllable second-order systems and on-chip digital-to-analog-converter-based bias generators, we show that the amplifier can be tuned in both bandwidth and center frequency while maintaining similar performance specifications to those achieved in the broadband mode of operation. In all cases, the power consumption of the amplifier is lower than 2.9mW. To the best of the authors' knowledge, this …

Sayan Das, Joseph C Bardin · 2021 IEEE MTT-S International Microwave Symposium (IMS), 892-895, 2021

Shot noise contributes significantly to the drain current noise in short-channel MOSFETs. The transport of carriers at the source-side of the channel is dominated by diffusion, leading to shot noise. However, channel resistance introduces carrier scattering, which eventually causes suppression of shot noise. The degree of suppression is described by a scaling factor, known as the Fano factor. This paper presents a detailed experimental evaluation of the Fano factor for nanometer-scale MOSFETs across seven different technology nodes, ranging from 12 nm to 180 nm. The dependence of the Fano factor on device bias, gate length, and channel doping is studied. We find that it is a strong function of device bias when the MOSFET operates at or close to the minimum noise bias. We also find that it rises for shorter gate length devices and is dependent on channel doping. The dependence of the Fano factor on channel …

Craig Gidney, N Cody Jones · arXiv preprint arXiv:2106.11513, 2021

We construct a CCCZ gate using six T gates, assisted by stabilizer operations and classical feedback. More generally, we reduce the T cost of a gate from to , for .

Google AI Quantum and Collaborators*†, Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B Buckley, David A Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J Huggins, Lev Ioffe, Sergei V Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V Klimov, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John M Martinis, Jarrod R McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E O’Brien, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C Rubin, Daniel Sank, Kevin J Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J Sung, Marco Szalay, Tyler Y Takeshita, Amit Vainsencher, Theodore White, Nathan Wiebe, Z Jamie Yao, Ping Yeh, Adam Zalcman · Science 369 (6507), 1084-1089, 2020

The simulation of fermionic systems is among the most anticipated applications of quantum computing. We performed several quantum simulations of chemistry with up to one dozen qubits, including modeling the isomerization mechanism of diazene. We also demonstrated error-mitigation strategies based on N-representability that dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realized the Givens rotation approach to noninteracting fermion evolution, which we variationally optimized to prepare the Hartree-Fock wave function. This ubiquitous algorithmic primitive is classically tractable to simulate yet still generates highly entangled states over the computational basis, which allowed us to assess the performance of our hardware and establish a foundation for scaling up correlated quantum chemistry simulations.

Jarrod R McClean, Nicholas C Rubin, Kevin J Sung, Ian D Kivlichan, Xavier Bonet-Monroig, Yudong Cao, Chengyu Dai, E Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, Thomas Häner, Tarini Hardikar, Vojtěch Havlíček, Oscar Higgott, Cupjin Huang, Josh Izaac, Zhang Jiang, Xinle Liu, Sam McArdle, Matthew Neeley, Thomas O’brien, Bryan O’gorman, Isil Ozfidan, Maxwell D Radin, Jhonathan Romero, Nicolas PD Sawaya, Bruno Senjean, Kanav Setia, Sukin Sim, Damian S Steiger, Mark Steudtner, Qiming Sun, Wei Sun, Daochen Wang, Fang Zhang, Ryan Babbush · Quantum Science and Technology 5 (3), 034014, 2020

Quantum simulation of chemistry and materials is predicted to be an important application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www. openfermion. org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic and bosonic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic …

Michael Broughton, Guillaume Verdon, Trevor McCourt, Antonio J Martinez, Jae Hyeon Yoo, Sergei V Isakov, Philip Massey, Ramin Halavati, Murphy Yuezhen Niu, Alexander Zlokapa, Evan Peters, Owen Lockwood, Andrea Skolik, Sofiene Jerbi, Vedran Dunjko, Martin Leib, Michael Streif, David Von Dollen, Hongxiang Chen, Shuxiang Cao, Roeland Wiersema, Hsin-Yuan Huang, Jarrod R McClean, Ryan Babbush, Sergio Boixo, Dave Bacon, Alan K Ho, Hartmut Neven, Masoud Mohseni · arXiv preprint arXiv:2003.02989, 2020

We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.

Brooks Foxen, Charles Neill, Andrew Dunsworth, Pedram Roushan, Ben Chiaro, Anthony Megrant, Julian Kelly, Zijun Chen, Kevin Satzinger, Rami Barends, F Arute, Kunal Arya, Ryan Babbush, Dave Bacon, JC Bardin, Sergio Boixo, D Buell, Brian Burkett, Yu Chen, Roberto Collins, Edward Farhi, Austin Fowler, C Gidney, Marissa Giustina, Rob Graff, M Harrigan, Trent Huang, SV Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Paul Klimov, Alexander Korotkov, Fedor Kostritsa, Dave Landhuis, Erik Lucero, Jarrod McClean, Matthew McEwen, Xiao Mi, Masoud Mohseni, JY Mutus, Ofer Naaman, Matthew Neeley, M Niu, A Petukhov, C Quintana, N Rubin, D Sank, V Smelyanskiy, A Vainsencher, TC White, Z Yao, P Yeh, A Zalcman, H Neven, John M Martinis, Google AI Quantum · Physical Review Letters 125 (12), 120504, 2020

Quantum algorithms offer a dramatic speedup for computational problems in material science and chemistry. However, any near-term realizations of these algorithms will need to be optimized to fit within the finite resources offered by existing noisy hardware. Here, taking advantage of the adjustable coupling of gmon qubits, we demonstrate a continuous two-qubit gate set that can provide a threefold reduction in circuit depth as compared to a standard decomposition. We implement two gate families: an imaginary swap-like (iSWAP-like) gate to attain an arbitrary swap angle, θ, and a controlled-phase gate that generates an arbitrary conditional phase, ϕ. Using one of each of these gates, we can perform an arbitrary two-qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic simulation (fSim) gate set. We benchmark the fidelity of the iSWAP-like and …

Benjamin Villalonga, Dmitry Lyakh, Sergio Boixo, Hartmut Neven, Travis S Humble, Rupak Biswas, Eleanor G Rieffel, Alan Ho, Salvatore Mandrà · Quantum Science and Technology 5 (3), 034003, 2020

Noisy intermediate-scale quantum (NISQ) computers are entering an era in which they can perform computational tasks beyond the capabilities of the most powerful classical computers, thereby achieving'quantum supremacy', a major milestone in quantum computing. NISQ supremacy requires comparison with a state-of-the-art classical simulator. We report HPC simulations of hard random quantum circuits (RQC), which have been recently used as a benchmark for the first experimental demonstration of quantum supremacy, sustaining an average performance of 281 Pflop/s (true single precision) on Summit, currently the fastest supercomputer in the world. These simulations were carried out using qFlex, a tensor-network-based classical high-performance simulator of RQCs. Our results show an advantage of many orders of magnitude in energy consumption of NISQ devices over classical supercomputers. In …

Tyler Takeshita, Nicholas C Rubin, Zhang Jiang, Eunseok Lee, Ryan Babbush, Jarrod R McClean · Physical Review X 10 (1), 011004, 2020

Proposals for experiments in quantum chemistry on quantum computers leverage the ability to target a subset of degrees of freedom containing the essential quantum behavior, sometimes called the active space. This approximation allows one to treat more difficult problems using fewer qubits and lower gate depths than would otherwise be possible. However, while this approximation captures many important qualitative features, it may leave the results wanting in terms of absolute accuracy (basis error) of the representation. In traditional approaches, increasing this accuracy requires increasing the number of qubits and an appropriate increase in circuit depth as well. Here we explore two techniques requiring no additional qubits or circuit depth that are able to remove much of this approximation in favor of additional measurements. The techniques are constructed and analyzed theoretically, and some numerical …

Ian D Kivlichan, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang, Nicholas Rubin, Austin Fowler, Alán Aspuru-Guzik, Hartmut Neven, Ryan Babbush · Quantum 4, 296, 2020

Recent work has deployed linear combinations of unitaries techniques to reduce the cost of fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve upon those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (eg energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with fermionic modes can be performed with roughly and T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; eg, we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.

Jarrod R McClean, Zhang Jiang, Nicholas C Rubin, Ryan Babbush, Hartmut Neven · Nature communications 11 (1), 636, 2020

With rapid developments in quantum hardware comes a push towards the first practical applications. While fully fault-tolerant quantum computers are not yet realized, there may exist intermediate forms of error correction that enable practical applications. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which mitigate errors on logical qubits using post-processing without explicit syndrome measurements or additional qubits beyond the encoding overhead. This greatly simplifies the experimental exploration of quantum codes on real, near-term devices, removing the need for locality of syndromes or fast feed-forward. We develop the theory of the method and demonstrate it on an example with the perfect [[5, 1, 3]] code, which exhibits a pseudo-threshold of p ≈ 0.50 under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration of …

Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Andreas Bengtsson, Sergio Boixo, Michael Broughton, Bob B Buckley, David A Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Yu-An Chen, Ben Chiaro, Roberto Collins, Stephen J Cotton, William Courtney, Sean Demura, Alan Derk, Andrew Dunsworth, Daniel Eppens, Thomas Eckl, Catherine Erickson, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Jonathan A Gross, Steve Habegger, Matthew P Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William Huggins, Lev B Ioffe, Sergei V Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V Klimov, Alexander N Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Michael Marthaler, Orion Martin, John M Martinis, Anika Marusczyk, Sam McArdle, Jarrod R McClean, Trevor McCourt, Matt McEwen, Anthony Megrant, Carlos Mejuto-Zaera, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Michael Newman, Murphy Yuezhen Niu, Thomas E O'Brien, Eric Ostby, Bálint Pató, Andre Petukhov, Harald Putterman, Chris Quintana, Jan-Michael Reiner, Pedram Roushan, Nicholas C Rubin, Daniel Sank, Kevin J Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J Sung, Peter Schmitteckert, Marco Szalay, Norm M Tubman, Amit Vainsencher, Theodore White, Nicolas Vogt, Z Jamie Yao, Ping Yeh, Adam Zalcman, Sebastian Zanker · arXiv preprint arXiv:2010.07965, 2020

Strongly correlated quantum systems give rise to many exotic physical phenomena, including high-temperature superconductivity. Simulating these systems on quantum computers may avoid the prohibitively high computational cost incurred in classical approaches. However, systematic errors and decoherence effects presented in current quantum devices make it difficult to achieve this. Here, we simulate the dynamics of the one-dimensional Fermi-Hubbard model using 16 qubits on a digital superconducting quantum processor. We observe separations in the spreading velocities of charge and spin densities in the highly excited regime, a regime that is beyond the conventional quasiparticle picture. To minimize systematic errors, we introduce an accurate gate calibration procedure that is fast enough to capture temporal drifts of the gate parameters. We also employ a sequence of error-mitigation techniques to reduce decoherence effects and residual systematic errors. These procedures allow us to simulate the time evolution of the model faithfully despite having over 600 two-qubit gates in our circuits. Our experiment charts a path to practical quantum simulation of strongly correlated phenomena using available quantum devices.

Edward Farhi, David Gamarnik, Sam Gutmann · arXiv preprint arXiv:2004.09002, 2020

The Quantum Approximate Optimization Algorithm can naturally be applied to combinatorial search problems on graphs. The quantum circuit has p applications of a unitary operator that respects the locality of the graph. On a graph with bounded degree, with p small enough, measurements of distant qubits in the state output by the QAOA give uncorrelated results. We focus on finding big independent sets in random graphs with dn/2 edges keeping d fixed and n large. Using the Overlap Gap Property of almost optimal independent sets in random graphs, and the locality of the QAOA, we are able to show that if p is less than a d-dependent constant times log n, the QAOA cannot do better than finding an independent set of size .854 times the optimal for d large. Because the logarithm is slowly growing, even at one million qubits we can only show that the algorithm is blocked if p is in single digits. At higher p the algorithm "sees" the whole graph and we have no indication that performance is limited.

Xavier Bonet-Monroig, Ryan Babbush, Thomas E O’Brien · Physical Review X 10 (3), 031064, 2020

Many applications of quantum simulation require one to prepare and then characterize quantum states by efficiently estimating k-body reduced density matrices (k-RDMs), from which observables of interest may be obtained. For instance, the fermionic 2-RDM contains the energy, charge density, and energy gradients of an electronic system, while the qubit 2-RDM contains the spatial correlation functions of magnetic systems. Naive estimation of such RDMs requires repeated state preparations for each matrix element, which makes for prohibitively large computation times. However, commuting matrix elements may be measured simultaneously, allowing for a significant cost reduction. In this work, we design schemes for such a parallelization with near-optimal complexity in the system size N. We first describe a scheme to sample all elements of a qubit k-RDM using only O (3 k log k− 1 N) unique measurement …

Kevin J Sung, Jiahao Yao, Matthew P Harrigan, Nicholas C Rubin, Zhang Jiang, Lin Lin, Ryan Babbush, Jarrod R McClean · Quantum Science and Technology 5 (4), 044008, 2020

Variational quantum algorithms are a leading candidate for early applications on noisy intermediate-scale quantum computers. These algorithms depend on a classical optimization outer-loop that minimizes some function of a parameterized quantum circuit. In practice, finite sampling error and gate errors make this a stochastic optimization with unique challenges that must be addressed at the level of the optimizer. The sharp trade-off between precision and sampling time in conjunction with experimental constraints necessitates the development of new optimization strategies to minimize overall wall clock time in this setting. In this work, we introduce two optimization methods and numerically compare their performance with common methods in use today. The methods are surrogate model-based algorithms designed to improve reuse of collected data. They do so by utilizing a least-squares quadratic fit of sampled …

Jacques Carolan, Masoud Mohseni, Jonathan P Olson, Mihika Prabhu, Changchen Chen, Darius Bunandar, Murphy Yuezhen Niu, Nicholas C Harris, Franco NC Wong, Michael Hochberg, Seth Lloyd, Dirk Englund · Nature Physics 16 (3), 322-327, 2020

A promising route towards the demonstration of near-term quantum advantage (or supremacy) over classical systems relies on running tailored quantum algorithms on noisy intermediate-scale quantum machines. These algorithms typically involve sampling from probability distributions that—under plausible complexity-theoretic conjectures—cannot be efficiently generated classically. Rather than determining the computational features of output states produced by a given physical system, we investigate what features of the generating system can be efficiently learnt given direct access to an output state. To tackle this question, here we introduce the variational quantum unsampling protocol, a nonlinear quantum neural network approach for verification and inference of near-term quantum circuit outputs. In our approach, one can variationally train a quantum operation to unravel the action of an unknown unitary on a …

Yuval R Sanders, Dominic W Berry, Pedro CS Costa, Louis W Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, Ryan Babbush · PRX Quantum 1 (2), 020312, 2020

Here we explore which heuristic quantum algorithms for combinatorial optimization might be most practical to try out on a small fault-tolerant quantum computer. We compile circuits for several variants of quantum-accelerated simulated annealing including those using qubitization or Szegedy walks to quantize classical Markov chains and those simulating spectral-gap-amplified Hamiltonians encoding a Gibbs state. We also optimize fault-tolerant realizations of the adiabatic algorithm, quantum-enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. Many of these methods are bottlenecked by calls to the same subroutines; thus, optimized circuits for those primitives should be of interest regardless of which heuristic is most effective in practice. We compile these bottlenecks for several families of optimization problems and report for how long and for what size systems …

Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven · Quantum 4, 276, 2020

We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an -mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than qubits on average. We apply it to the problem of learning -fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all -fermion RDMs in parallel, to precision , by repeating a single quantum circuit for times. This result is based on a method we develop here that allows one to determine individual elements of all -qubit RDMs in parallel, to precision , by repeating a single quantum circuit for times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

Vadim N Smelyanskiy, Kostyantyn Kechedzhi, Sergio Boixo, Sergei V Isakov, Hartmut Neven, Boris Altshuler · Physical Review X 10 (1), 011017, 2020

We address the long-standing problem of the structure of the low-energy eigenstates and long-time coherent dynamics in quantum spin-glass models. Below the spin-glass freezing transition, the energy landscape of the spin system is characterized by a proliferation of local minima where classical dynamics gets trapped. A theoretical description of quantum dynamics in this regime is challenging due to the complex nature of the distribution of the tunneling matrix elements between the local minima of the energy landscape. We study the transverse-field-induced quantum dynamics of the following “impurity band”(IB) spin model: zero energy of all spin configurations except for a small fraction of spin configurations (“marked states”) that form a narrow band at a large negative energy. At a zero transverse field, the IB model demonstrates the freezing transition at inverse temperature β f∼ 1 characterized by a nonzero …

Paul V Klimov, Julian Kelly, John M Martinis, Hartmut Neven · arXiv preprint arXiv:2006.04594, 2020

High performance quantum computing requires a calibration system that learns optimal control parameters much faster than system drift. In some cases, the learning procedure requires solving complex optimization problems that are non-convex, high-dimensional, highly constrained, and have astronomical search spaces. Such problems pose an obstacle for scalability since traditional global optimizers are often too inefficient and slow for even small-scale processors comprising tens of qubits. In this whitepaper, we introduce the Snake Optimizer for efficiently and quickly solving such optimization problems by leveraging concepts in artificial intelligence, dynamic programming, and graph optimization. In practice, the Snake has been applied to optimize the frequencies at which quantum logic gates are implemented in frequency-tunable superconducting qubits. This application enabled state-of-the-art system performance on a 53 qubit quantum processor, serving as a key component of demonstrating quantum supremacy. Furthermore, the Snake Optimizer scales favorably with qubit number and is amenable to both local re-optimization and parallelization, showing promise for optimizing much larger quantum processors.

Jarrod R McClean, Fabian M Faulstich, Qinyi Zhu, Bryan O’Gorman, Yiheng Qiu, Steven R White, Ryan Babbush, Lin Lin · New Journal of Physics 22 (9), 093015, 2020

All-electron electronic structure methods based on the linear combination of atomic orbitals method with Gaussian basis set discretization offer a well established, compact representation that forms much of the foundation of modern correlated quantum chemistry calculations—on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block-diagonal …

Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P Harrigan, Michael J Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S Humble, Sergei V Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C Platt, Chris Quintana, Eleanor G Rieffel, Pedram Roushan, Nicholas C Rubin, Daniel Sank, Kevin J Satzinger, Vadim Smelyanskiy, Kevin J Sung, Matthew D Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, John M Martinis · Nature 574 (7779), 505-510, 2019

The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor 1. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits 2, 3, 4, 5, 6, 7 to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 2 53 (about 10 16). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times—our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This …

Murphy Yuezhen Niu, Sergio Boixo, Vadim N Smelyanskiy, Hartmut Neven · npj Quantum Information 5 (1), 33, 2019

Emerging reinforcement learning techniques using deep neural networks have shown great promise in control optimization. They harness non-local regularities of noisy control trajectories and facilitate transfer learning between tasks. To leverage these powerful capabilities for quantum control optimization, we propose a new control framework to simultaneously optimize the speed and fidelity of quantum computation against both leakage and stochastic control errors. For a broad family of two-qubit unitary gates that are important for quantum simulation of many-electron systems, we improve the control robustness by adding control noise into training environments for reinforcement learning agents trained with trusted-region-policy-optimization. The agent control solutions demonstrate a two-order-of-magnitude reduction in average-gate-error over baseline stochastic-gradient-descent solutions and up to a one-order …

Guillaume Verdon, Michael Broughton, Jarrod R McClean, Kevin J Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, Masoud Mohseni · arXiv preprint arXiv:1907.05415, 2019

Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.

Dominic W Berry, Craig Gidney, Mario Motta, Jarrod R McClean, Ryan Babbush · Quantum 3, 208, 2019

Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with T complexity, where is number of orbitals and is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.

Benjamin Villalonga, Sergio Boixo, Bron Nelson, Christopher Henze, Eleanor Rieffel, Rupak Biswas, Salvatore Mandrà · npj Quantum Information 5 (1), 86, 2019

Here we present qFlex, a flexible tensor network-based quantum circuit simulator. qFlex can compute both the exact amplitudes, essential for the verification of the quantum hardware, as well as low-fidelity amplitudes, to mimic sampling from Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we focus on random quantum circuits (RQCs) in the range of sizes expected for supremacy experiments. Fidelity f simulations are performed at a cost that is 1/f lower than perfect fidelity ones. We also present a technique to eliminate the overhead introduced by rejection sampling in most tensor network approaches. We benchmark the simulation of square lattices and Google’s Bristlecone QPU. Our analysis is supported by extensive simulations on NASA HPC clusters Pleiades and Electra. For our most computationally demanding simulation, the two clusters combined reached a peak of 20 Peta Floating Point …

Joseph C Bardin, Evan Jeffrey, Erik Lucero, Trent Huang, Sayan Das, Daniel Thomas Sank, Ofer Naaman, Anthony Edward Megrant, Rami Barends, Ted White, Marissa Giustina, Kevin J Satzinger, Kunal Arya, Pedram Roushan, Benjamin Chiaro, Julian Kelly, Zijun Chen, Brian Burkett, Yu Chen, Andrew Dunsworth, Austin Fowler, Brooks Foxen, Craig Gidney, Rob Graff, Paul Klimov, Josh Mutus, Matthew J McEwen, Matthew Neeley, Charles J Neill, Chris Quintana, Amit Vainsencher, Hartmut Neven, John Martinis · IEEE Journal of Solid-State Circuits 54 (11), 3043-3060, 2019

Implementation of an error-corrected quantum computer is believed to require a quantum processor with a million or more physical qubits, and, in order to run such a processor, a quantum control system of similar scale will be required. Such a controller will need to be integrated within the cryogenic system and in close proximity with the quantum processor in order to make such a system practical. Here, we present a prototype cryogenic CMOS quantum controller designed in a 28-nm bulk CMOS process and optimized to implement a 16-word (4-bit) XY gate instruction set for controlling transmon qubits. After introducing the transmon qubit, including a discussion of how it is controlled, design considerations are discussed, with an emphasis on error rates and scalability. The circuit design is then discussed. Cryogenic performance of the underlying technology is presented, and the results of several quantum control …

Craig Gidney, Austin G Fowler · Quantum 3, 135, 2019

We present magic state factory constructions for producing states and states. For the factory we apply the surface code lattice surgery construction techniques described in [15] to the fault-tolerant Toffoli [21, 12]. The resulting factory has a footprint of (where is the code distance) and produces one every surface code cycles. Our state factory uses the factory's output and a catalyst state to exactly transform one state into two states. It has a footprint smaller than the factory in [15] but outputs states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the case produces a particularly efficient circuit for producing states. Compared to using the factory of [15], our factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm [31] and the chemistry algorithm of Babbush et al.[1]. Assuming a physical gate error rate of , our CCZ factory can produce states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to by increasing the factory footprint .

Rami Barends, CM Quintana, AG Petukhov, Yu Chen, Dvir Kafri, Kostyantyn Kechedzhi, Roberto Collins, Ofer Naaman, Sergio Boixo, F Arute, Kunal Arya, D Buell, Brian Burkett, Z Chen, Ben Chiaro, Andrew Dunsworth, Brooks Foxen, Austin Fowler, C Gidney, Marissa Giustina, Rob Graff, Trent Huang, Evan Jeffrey, J Kelly, Paul Victor Klimov, Fedor Kostritsa, Dave Landhuis, Erik Lucero, Matt McEwen, Anthony Megrant, Xiao Mi, Josh Mutus, Matthew Neeley, Charles Neill, Eric Ostby, Pedram Roushan, Daniel Sank, KJ Satzinger, A Vainsencher, T White, J Yao, P Yeh, A Zalcman, H Neven, VN Smelyanskiy, John M Martinis · Physical review letters 123 (21), 210501, 2019

We demonstrate diabatic two-qubit gates with Pauli error rates down to 4.3 (2)× 10− 3 in as fast as 18 ns using frequency-tunable superconducting qubits. This is achieved by synchronizing the entangling parameters with minima in the leakage channel. The synchronization shows a landscape in gate parameter space that agrees with model predictions and facilitates robust tune-up. We test both i swap-like and cphase gates with cross-entropy benchmarking. The presented approach can be extended to multibody operations as well.

Ryan Babbush, Dominic W Berry, Hartmut Neven · Physical Review A 99 (4), 040301, 2019

We show that one can quantum simulate the dynamics of a Sachdev-Ye-Kitaev model with N Majorana modes for time t to precision ε with gate complexity O (N 7/2 t+ N 5/2 t polylog (N/ε)). In addition to scaling sublinearly in the number of Hamiltonian terms, this gate complexity represents an exponential improvement in 1/ε and large polynomial improvement in N and t over prior state-of-the-art algorithms which scale as O (N 10 t 2/ε). Our approach involves a variant of the qubitization technique in which we encode the Hamiltonian H as an asymmetric projection of a signal oracle U onto two different signal states prepared by state oracles, A| 0〉↦| A〉 and B| 0〉↦| B〉, such that H=〈 B| U| A〉. Our strategy for applying this method to the Sachdev-Ye-Kitaev model involves realizing B using only Hadamard gates and realizing A as a random quantum circuit.

Ryan Babbush, Dominic W Berry, Jarrod R McClean, Hartmut Neven · npj Quantum Information 5 (1), 92, 2019

We present a quantum algorithm for simulating quantum chemistry with gate complexity where η is the number of electrons and N is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity . We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when N ≫ η, as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation.

Zhang Jiang, Jarrod McClean, Ryan Babbush, Hartmut Neven · Physical Review Applied 12 (6), 064041, 2019

Fermion-to-qubit mappings that preserve geometric locality are especially useful for simulating lattice fermion models (eg, the Hubbard model) on a quantum computer. They avoid the overhead associated with geometric nonlocal parity terms in mappings such as the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. As a result, they often provide quantum circuits with lower depth and gate complexity. In such encodings, fermionic states are encoded in the common+ 1 eigenspace of a set of stabilizers, akin to stabilizer quantum error-correcting codes. Here, we discuss several known geometric locality-preserving mappings and their abilities to correct and detect single-qubit errors. We introduce a geometric locality-preserving map, whose stabilizers correspond to products of Majorana operators on closed paths of the fermionic hopping graph. We show that our code, which we refer to as the …

Craig Gidney, Austin G Fowler · arXiv preprint arXiv:1905.08916, 2019

We construct a self-correcting CCZ state (the "AutoCCZ") with embedded delayed choice CZs for completing gate teleportations. Using the AutoCCZ state we create efficient surface code spacetime layouts for both a depth-limited circuit (a ripply-carry addition) and a Clifford-limited circuit (a QROM read). Our layouts account for distillation and routing, are based on plausible physical assumptions for a large-scale superconducting qubit platform, and suggest that circuit-level Toffoli parallelism (e.g. using a carry-lookahead adder instead of a ripple-carry adder) will not reduce the execution time of computations involving fewer than five million physical qubits. We reduce the spacetime volume of delayed choice CZs by a factor of 4 compared to techniques from previous work (Fowler 2012), and make several improvements to the CCZ magic state factory from (Gidney 2019).

James King, Masoud Mohseni, William Bernoudy, Alexandre Fréchette, Hossein Sadeghi, Sergei V Isakov, Hartmut Neven, Mohammad H Amin · arXiv preprint arXiv:1907.00707, 2019

Genetic algorithms, which mimic evolutionary processes to solve optimization problems, can be enhanced by using powerful semi-local search algorithms as mutation operators. Here, we introduce reverse quantum annealing, a class of quantum evolutions that can be used for performing families of quasi-local or quasi-nonlocal search starting from a classical state, as novel sources of mutations. Reverse annealing enables the development of genetic algorithms that use quantum fluctuation for mutations and classical mechanisms for the crossovers -- we refer to these as Quantum-Assisted Genetic Algorithms (QAGAs). We describe a QAGA and present experimental results using a D-Wave 2000Q quantum annealing processor. On a set of spin-glass inputs, standard (forward) quantum annealing finds good solutions very quickly but struggles to find global optima. In contrast, our QAGA proves effective at finding global optima for these inputs. This successful interplay of non-local classical and quantum fluctuations could provide a promising step toward practical applications of Noisy Intermediate-Scale Quantum (NISQ) devices for heuristic discrete optimization.

Craig Gidney · arXiv preprint arXiv:1904.07356, 2019

We improve the space complexity of Karatsuba multiplication on a quantum computer from to while maintaining gate complexity. We achieve this by ensuring recursive calls can add their outputs directly into subsections of the output register. This avoids the need to store, and uncompute, intermediate results. This optimization, which is analogous to classical tail-call optimization, should be applicable to a wide range of recursive quantum algorithms.

Craig Gidney · arXiv preprint arXiv:1905.07682, 2019

We demonstrate a technique for optimizing quantum circuits that is analogous to classical windowing. Specifically, we show that small table lookups can allow control qubits to be iterated in groups instead of individually. We present various windowed quantum arithmetic circuits, including a windowed modular exponentiation with nested windowed modular multiplications, which have lower Toffoli counts than previous work at register sizes ranging from tens of qubits to thousands of qubits.

Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, Hartmut Neven · Nature communications 9 (1), 4812, 2018

Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design …

Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J Bremner, John M Martinis, Hartmut Neven · Nature Physics 14 (6), 595-600, 2018

A critical question for quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of supercomputers. Such a demonstration of what is referred to as quantum supremacy requires a reliable evaluation of the resources required to solve tasks with classical approaches. Here, we propose the task of sampling from the output distribution of random quantum circuits as a demonstration of quantum supremacy. We extend previous results in computational complexity to argue that this sampling task must take exponential time in a classical computer. We introduce cross-entropy benchmarking to obtain the experimental fidelity of complex multiqubit dynamics. This can be estimated and extrapolated to give a success metric for a quantum supremacy demonstration. We study the computational cost of relevant classical …

Edward Farhi, Hartmut Neven · arXiv preprint arXiv:1802.06002, 2018

We introduce a quantum neural network, QNN, that can represent labeled data, classical or quantum, and be trained by supervised learning. The quantum circuit consists of a sequence of parameter dependent unitary transformations which acts on an input quantum state. For binary classification a single Pauli operator is measured on a designated readout qubit. The measured output is the quantum neural network's predictor of the binary label of the input state. First we look at classifying classical data sets which consist of n-bit strings with binary labels. The input quantum state is an n-bit computational basis state corresponding to a sample string. We show how to design a circuit made from two qubit unitaries that can correctly represent the label of any Boolean function of n bits. For certain label functions the circuit is exponentially long. We introduce parameter dependent unitaries that can be adapted by supervised learning of labeled data. We study an example of real world data consisting of downsampled images of handwritten digits each of which has been labeled as one of two distinct digits. We show through classical simulation that parameters can be found that allow the QNN to learn to correctly distinguish the two data sets. We then discuss presenting the data as quantum superpositions of computational basis states corresponding to different label values. Here we show through simulation that learning is possible. We consider using our QNN to learn the label of a general quantum state. By example we show that this can be done. Our work is exploratory and relies on the classical simulation of small quantum systems. The QNN proposed …

Cornelius Hempel, Christine Maier, Jonathan Romero, Jarrod McClean, Thomas Monz, Heng Shen, Petar Jurcevic, Ben P Lanyon, Peter Love, Ryan Babbush, Alán Aspuru-Guzik, Rainer Blatt, Christian F Roos · Physical Review X 8 (3), 031022, 2018

Quantum-classical hybrid algorithms are emerging as promising candidates for near-term practical applications of quantum information processors in a wide variety of fields ranging from chemistry to physics and materials science. We report on the experimental implementation of such an algorithm to solve a quantum chemistry problem, using a digital quantum simulator based on trapped ions. Specifically, we implement the variational quantum eigensolver algorithm to calculate the molecular ground-state energies of two simple molecules and experimentally demonstrate and compare different encoding methods using up to four qubits. Furthermore, we discuss the impact of measurement noise as well as mitigation strategies and indicate the potential for adaptive implementations focused on reaching chemical accuracy, which may serve as a cross-platform benchmark for multiqubit quantum simulators.

Jonathan Romero, Ryan Babbush, Jarrod R McClean, Cornelius Hempel, Peter J Love, Alán Aspuru-Guzik · Quantum Science and Technology 4 (1), 014008, 2018

The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the …

Charles Neill, Pedran Roushan, K Kechedzhi, Sergio Boixo, Sergei V Isakov, V Smelyanskiy, A Megrant, B Chiaro, A Dunsworth, K Arya, Rami Barends, B Burkett, Y Chen, Z Chen, A Fowler, B Foxen, M Giustina, R Graff, E Jeffrey, T Huang, J Kelly, P Klimov, E Lucero, J Mutus, M Neeley, C Quintana, D Sank, A Vainsencher, J Wenner, TC White, Hartmut Neven, John M Martinis · Science 360 (6385), 195-199, 2018

A key step toward demonstrating a quantum system that can address difficult problems in physics and chemistry will be performing a computation beyond the capabilities of any classical computer, thus achieving so-called quantum supremacy. In this study, we used nine superconducting qubits to demonstrate a promising path toward quantum supremacy. By individually tuning the qubit parameters, we were able to generate thousands of distinct Hamiltonian evolutions and probe the output probabilities. The measured probabilities obey a universal distribution, consistent with uniformly sampling the full Hilbert space. As the number of qubits increases, the system continues to explore the exponentially growing number of states. Extending these results to a system of 50 qubits has the potential to address scientific questions that are beyond the capabilities of any classical computer.

Ian D Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, Ryan Babbush · Physical review letters 120 (11), 110501, 2018

As physical implementations of quantum architectures emerge, it is increasingly important to consider the cost of algorithms for practical connectivities between qubits. We show that by using an arrangement of gates that we term the fermionic swap network, we can simulate a Trotter step of the electronic structure Hamiltonian in exactly N depth and with N 2/2 two-qubit entangling gates, and prepare arbitrary Slater determinants in at most N/2 depth, all assuming only a minimal, linearly connected architecture. We conjecture that no explicit Trotter step of the electronic structure Hamiltonian is possible with fewer entangling gates, even with arbitrary connectivities. These results represent significant practical improvements on the cost of most Trotter-based algorithms for both variational and phase-estimation-based simulation of quantum chemistry.

Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, Garnet Kin-Lic Chan · Physical Review X 8 (1), 011044, 2018

Quantum simulation of the electronic structure problem is one of the most researched applications of quantum computing. The majority of quantum algorithms for this problem encode the wavefunction using N Gaussian orbitals, leading to Hamiltonians with O (N 4) second-quantized terms. We avoid this overhead and extend methods to condensed phase materials by utilizing a dual form of the plane wave basis which diagonalizes the potential operator, leading to a Hamiltonian representation with O (N 2) second-quantized terms. Using this representation, we can implement single Trotter steps of the Hamiltonians with linear gate depth on a planar lattice. Properties of the basis allow us to deploy Trotter-and Taylor-series-based simulations with respective circuit depths of O (N 7/2) and O (N 8/3) for fixed charge densities. Variational algorithms also require significantly fewer measurements in this basis, ameliorating …

Ryan Babbush, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, Hartmut Neven · Physical Review X 8 (4), 041015, 2018

We construct quantum circuits that exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced “qubitization” framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity O (λ/ε), where λ is an absolute sum of Hamiltonian coefficients and ε is the target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T-gate complexity O (N+ log (1/ε)), where N is the number of orbitals in the basis. This scenario enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O (N 3/ε+ N 2 log (1/ε)/ε). Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the …

PV Klimov, Julian Kelly, Zijun Chen, Matthew Neeley, Anthony Megrant, Brian Burkett, Rami Barends, Kunal Arya, Ben Chiaro, Yu Chen, Andrew Dunsworth, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Trent Huang, Evan Jeffrey, Erik Lucero, Josh Y Mutus, Ofer Naaman, Charles Neill, Chris Quintana, Pedram Roushan, Daniel Sank, Amit Vainsencher, Jim Wenner, Timothy C White, Sergio Boixo, Ryan Babbush, Vadim N Smelyanskiy, Hartmut Neven, John M Martinis · Physical review letters 121 (9), 090502, 2018

Superconducting qubits are an attractive platform for quantum computing since they have demonstrated high-fidelity quantum gates and extensibility to modest system sizes. Nonetheless, an outstanding challenge is stabilizing their energy-relaxation times, which can fluctuate unpredictably in frequency and time. Here, we use qubits as spectral and temporal probes of individual two-level-system defects to provide direct evidence that they are responsible for the largest fluctuations. This research lays the foundation for stabilizing qubit performance through calibration, design, and fabrication.

Craig Gidney · Quantum 2, 74, 2018

We improve the number of T gates needed to perform an n-bit adder from to . We do so via a" temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of , a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.

Raffaele Santagati, Jianwei Wang, Antonio A Gentile, Stefano Paesani, Nathan Wiebe, Jarrod R McClean, Sam Morley-Short, Peter J Shadbolt, Damien Bonneau, Joshua W Silverstone, David P Tew, Xiaoqi Zhou, Jeremy L O’Brien, Mark G Thompson · Science advances 4 (1), eaap9646, 2018

The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. We introduce the concept of an “eigenstate witness” and, through it, provide a new quantum approach that combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32 bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result …

Zhang Jiang, Kevin J Sung, Kostyantyn Kechedzhi, Vadim N Smelyanskiy, Sergio Boixo · Physical Review Applied 9 (4), 044036, 2018

Simulating strongly correlated fermionic systems is notoriously hard on classical computers. An alternative approach, as proposed by Feynman, is to use a quantum computer. We discuss simulating strongly correlated fermionic systems using near-term quantum devices. We focus specifically on two-dimensional (2D) or linear geometry with nearest-neighbor qubit-qubit couplings, typical for superconducting transmon qubit arrays. We improve an existing algorithm to prepare an arbitrary Slater determinant by exploiting a unitary symmetry. We also present a quantum algorithm to prepare an arbitrary fermionic Gaussian state with O (N 2) gates and O (N) circuit depth. Both algorithms are optimal in the sense that the numbers of parameters in the quantum circuits are equal to those describing the quantum states. Furthermore, we propose an algorithm to implement the 2D fermionic Fourier transformation on a 2D qubit …

Nicholas C Rubin, Ryan Babbush, Jarrod McClean · New Journal of Physics 20 (5), 053020, 2018

Many quantum algorithms, including recently proposed hybrid classical/quantum algorithms, make use of restricted tomography of the quantum state that measures the reduced density matrices, or marginals, of the full state. The most straightforward approach to this algorithmic step estimates each component of the marginal independently without making use of the algebraic and geometric structure of the marginals. Within the field of quantum chemistry, this structure is termed the fermionic n-representability conditions, and is supported by a vast amount of literature on both theoretical and practical results related to their approximations. In this work, we introduce these conditions in the language of quantum computation, and utilize them to develop several techniques to accelerate and improve practical applications for quantum chemistry on quantum computers. As a general result, we demonstrate how these …

Fernando GSL Brandao, Michael Broughton, Edward Farhi, Sam Gutmann, Hartmut Neven · arXiv preprint arXiv:1812.04170, 2018

The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth quantum circuit to produce a parameter dependent state. For a given combinatorial optimization problem instance, the quantum expectation of the associated cost function is the parameter dependent objective function of the QAOA. We demonstrate that if the parameters are fixed and the instance comes from a reasonable distribution then the objective function value is concentrated in the sense that typical instances have (nearly) the same value of the objective function. This applies not just for optimal parameters as the whole landscape is instance independent. We can prove this is true for low depth quantum circuits for instances of MaxCut on large 3-regular graphs. Our results generalize beyond this example. We support the arguments with numerical examples that show remarkable concentration. For higher depth circuits the numerics also show concentration and we argue for this using the Law of Large Numbers. We also observe by simulation that if we find parameters which result in good performance at say 10 bits these same parameters result in good performance at say 24 bits. These findings suggest ways to run the QAOA that reduce or eliminate the use of the outer loop optimization and may allow us to find good solutions with fewer calls to the quantum computer.

Igor L Markov, Aneeqa Fatima, Sergei V Isakov, Sergio Boixo · arXiv preprint arXiv:1807.10749, 2018

As quantum computers improve in the number of qubits and fidelity, the question of when they surpass state-of-the-art classical computation for a well-defined computational task is attracting much attention. The leading candidate task for this milestone entails sampling from the output distribution defined by a random quantum circuit. We develop a massively-parallel simulation tool Rollright that does not require inter-process communication (IPC) or proprietary hardware. We also develop two ways to trade circuit fidelity for computational speedups, so as to match the fidelity of a given quantum computer --- a task previously thought impossible. We report massive speedups for the sampling task over prior software from Microsoft, IBM, Alibaba and Google, as well as supercomputer and GPU-based simulations. By using publicly available Google Cloud Computing, we price such simulations and enable comparisons by total cost across hardware platforms. We simulate approximate sampling from the output of a circuit with 7x8 qubits and depth 1+40+1 by producing one million bitstring probabilities with fidelity 0.5%, at an estimated cost of $35184. The simulation costs scale linearly with fidelity, and using this scaling we estimate that extending circuit depth to 1+48+1 increases costs to one million dollars. Scaling the simulation to 10M bitstring probabilities needed for sampling 1M bitstrings helps comparing simulation to quantum computers. We describe refinements in benchmarks that slow down leading simulators, halving the circuit depth that can be simulated within the same time.

Austin G Fowler, Craig Gidney · arXiv preprint arXiv:1808.06709, 2018

When calculating the overhead of a quantum algorithm made fault-tolerant using the surface code, many previous works have used defects and braids for logical qubit storage and state distillation. In this work, we show that lattice surgery reduces the storage overhead by over a factor of 4, and the distillation overhead by nearly a factor of 5, making it possible to run algorithms with T gates using only physical qubits capable of executing gates with error . These numbers strongly suggest that defects and braids in the surface code should be deprecated in favor of lattice surgery.

Dominic W Berry, Mária Kieferová, Artur Scherer, Yuval R Sanders, Guang Hao Low, Nathan Wiebe, Craig Gidney, Ryan Babbush · npj Quantum Information 4 (1), 22, 2018

Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for …

Norm M Tubman, Carlos Mejuto-Zaera, Jeffrey M Epstein, Diptarka Hait, Daniel S Levine, William Huggins, Zhang Jiang, Jarrod R McClean, Ryan Babbush, Martin Head-Gordon, K Birgitta Whaley · arXiv preprint arXiv:1809.05523, 2018

Despite significant work on resource estimation for quantum simulation of electronic systems, the challenge of preparing states with sufficient ground state support has so far been largely neglected. In this work we investigate this issue in several systems of interest, including organic molecules, transition metal complexes, the uniform electron gas, Hubbard models, and quantum impurity models arising from embedding formalisms such as dynamical mean-field theory. Our approach uses a state-of-the-art classical technique for high-fidelity ground state approximation. We find that easy-to-prepare single Slater determinants such as the Hartree-Fock state often have surprisingly robust support on the ground state for many applications of interest. For the most difficult systems, single-determinant reference states may be insufficient, but low-complexity reference states may suffice. For this we introduce a method for preparation of multi-determinant states on quantum computers.

Kostyantyn Kechedzhi, Vadim Smelyanskiy, Jarrod R McClean, Vasil S Denchev, Masoud Mohseni, Sergei Isakov, Sergio Boixo, Boris Altshuler, Hartmut Neven · arXiv preprint arXiv:1807.04792, 2018

We analyze a new computational role of coherent multi-qubit quantum tunneling that gives rise to bands of non-ergodic extended (NEE) quantum states each formed by a superposition of a large number of computational states (deep local minima of the energy landscape) with similar energies. NEE provide a mechanism for population transfer (PT) between computational states and therefore can serve as a new quantum subroutine for quantum search, quantum parallel tempering and reverse annealing optimization algorithms. We study PT in a quantum n-spin system subject to a transverse field where the energy function encodes a classical optimization problem over the set of spin configurations . Given an initial spin configuration with low energy, PT protocol searches for other bitstrings at energies within a narrow window around the initial one. We provide an analytical solution for PT in a simple yet nontrivial model: randomly chosen marked bit-strings are assigned energies within a narrow strip , while the rest of the states are assigned energy 0. We find that the scaling of a typical PT runtime with n and L is the same as that in the multi-target Grover's quantum search algorithm, except for a factor that is equal to for finite transverse field . Unlike the Hamiltonians used in analog quantum unstructured search algorithms known so far, the model we consider is non-integrable and population transfer is not exponentially sensitive in n to the weight of the driver Hamiltonian. We study numerically the PT subroutine as a part of quantum parallel tempering algorithm for a number of examples of binary …

Alan Ho, Jarrod McClean, Shyue Ping Ong · Joule 2 (5), 810-813, 2018

Jarrod McClean is a research scientist in Google's Quantum Artificial Intelligence Lab working on the development of practical quantum algorithms for quantum simulation and other problems. He received his PhD in Chemical Physics from Harvard University specializing in quantum chemistry and quantum computation supported by the US Department of Energy as a Computational Science Graduate Fellow. His research interests broadly include quantum computation, quantum chemistry, numerical methods, and information sparsity.Alan Ho is a product manager in Google's Quantum Artificial Intelligence Lab working on identifying applications of quantum computing. He has spent his career in a number of engineering and entrepreneurial roles. His engineering interests broadly include quantum computation, quantum chemistry, and productization of new technologies.Shyue Ping Ong is an Associate Professor in …

Masoud Mohseni, Johan Strumpfer, Marek M Rams · New Journal of Physics 20 (10), 105002, 2018

We introduce a phenomenological theory for many-body control of critical phenomena by engineering causally-induced gaps for quantum Hamiltonian systems. The core mechanisms are controlling information flow within and/or between clusters that are created near a quantum critical point. To this end, we construct inhomogeneous quantum phase transitions via designing spatiotemporal quantum fluctuations. We show how non-equilibrium evolution of disordered quantum systems can create new effective correlation length scales and effective dynamical critical exponents. In particular, we construct a class of causally-induced non-adiabatic quantum annealing transitions for strongly disordered quantum Ising chains leading to exponential suppression of topological defects beyond standard Kibble–Zurek predictions. Using exact numerical techniques for 1D quantum Hamiltonian systems, we demonstrate that our …

Pedram Roushan, Charles Neill, J Tangpanitanon, Victor M Bastidas, A Megrant, Rami Barends, Yu Chen, Z Chen, B Chiaro, A Dunsworth, A Fowler, B Foxen, Marissa Giustina, E Jeffrey, J Kelly, Erik Lucero, J Mutus, Matthew Neeley, Chris Quintana, D Sank, Amit Vainsencher, James Wenner, T White, H Neven, DG Angelakis, J Martinis · Science 358 (6367), 1175-1179, 2017

Quantized eigenenergies and their associated wave functions provide extensive information for predicting the physics of quantum many-body systems. Using a chain of nine superconducting qubits, we implement a technique for resolving the energy levels of interacting photons. We benchmark this method by capturing the main features of the intricate energy spectrum predicted for two-dimensional electrons in a magnetic field—the Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels of the system as it undergoes the transition from a thermalized to a localized phase. Our work introduces a many-body spectroscopy technique to study quantum phases of matter.

Pedram Roushan, Charles Neill, Anthony Megrant, Yu Chen, Ryan Babbush, Rami Barends, Brooks Campbell, Zijun Chen, Ben Chiaro, Andrew Dunsworth, Austin Fowler, Evan Jeffrey, Julian Kelly, Erik Lucero, Josh Mutus, PJJ O’Malley, Matthew Neeley, Chris Quintana, Daniel Sank, Amit Vainsencher, Jim Wenner, Ted White, Eliot Kapit, Hartmut Neven, John Martinis · Nature Physics 13 (2), 146-151, 2017

The intriguing many-body phases of quantum matter arise from the interplay of particle interactions, spatial symmetries, and external fields. Generating these phases in an engineered system could provide deeper insight into their nature. Using superconducting qubits, we simultaneously realize synthetic magnetic fields and strong particle interactions, which are among the essential elements for studying quantum magnetism and fractional quantum Hall phenomena. The artificial magnetic fields are synthesized by sinusoidally modulating the qubit couplings. In a closed loop formed by the three qubits, we observe the directional circulation of photons, a signature of broken time-reversal symmetry. We demonstrate strong interactions through the creation of photon vacancies, or ‘holes’, which circulate in the opposite direction. The combination of these key elements results in chiral ground-state currents. Our work …

Bjoern Lekitsch, Sebastian Weidt, Austin G Fowler, Klaus Mølmer, Simon J Devitt, Christof Wunderlich, Winfried K Hensinger · Science Advances 3 (2), e1601540, 2017

The availability of a universal quantum computer may have a fundamental impact on a vast number of research fields and on society as a whole. An increasingly large scientific and industrial community is working toward the realization of such a device. An arbitrarily large quantum computer may best be constructed using a modular approach. We present a blueprint for a trapped ion–based scalable quantum computer module, making it possible to create a scalable quantum computer architecture based on long-wavelength radiation quantum gates. The modules control all operations as stand-alone units, are constructed using silicon microfabrication techniques, and are within reach of current technology. To perform the required quantum computations, the modules make use of long-wavelength radiation–based quantum gate technology. To scale this microwave quantum computer architecture to a large size, we …

Masoud Mohseni, Peter Read, Hartmut Neven, Sergio Boixo, Vasil Denchev, Ryan Babbush, Austin Fowler, Vadim Smelyanskiy, John Martinis · Nature 543 (7644), 171-174, 2017

Masoud Mohseni, Peter Read, Hartmut Neven and colleagues at Google's Quantum AI Laboratory set out investment opportunities on the road to the ultimate quantum machines.

Zhi-Cheng Yang, Armin Rahmani, Alireza Shabani, Hartmut Neven, Claudio Chamon · Physical Review X 7 (2), 021027, 2017

We use Pontryagin’s minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed quantum approximate optimization algorithm. Focusing on the Sherrington-Kirkpatrick spin glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parametrization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. Furthermore, we find that the success rates …

Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Hartmut Neven · arXiv preprint arXiv:1712.05384, 2017

Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond the capabilities of known classical algorithms on state-of-the-art computers, achieving the first milestone of so-called quantum supremacy. This has stimulated recent progress in classical algorithms to simulate quantum circuits. Classical simulations are also necessary to approximate the fidelity of multiqubit quantum computers using cross entropy benchmarking. Here we present numerical results of a classical simulation algorithm to sample universal random circuits, on a single workstation, with more qubits and depth than previously reported. For example, circuits with qubits of depth 37, qubits of depth 27, and ) qubits of depth 19 are all easy to sample. We also show up to what depth the sampling, or estimation of observables, is trivially parallelizable. The algorithm is related to the "Feynmann path" method to simulate quantum circuits. For low-depth circuits, the algorithm scales exponentially in the depth times the smaller lateral dimension, or the treewidth, as explained in Boixo et. al., and therefore confirms the bounds in that paper. In particular, circuits with qubits and depth 40 remain currently out of reach. Follow up work on a supercomputer environment will tighten this bound.

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Hartmut Neven · arXiv preprint arXiv:1703.06199, 2017

Gate model quantum computers with too many qubits to be simulated by available classical computers are about to arrive. We present a strategy for programming these devices without error correction or compilation. This means that the number of logical qubits is the same as the number of qubits on the device. The hardware determines which pairs of qubits can be addressed by unitary operators. The goal is to build quantum states that solve computational problems such as maximizing a combinatorial objective function or minimizing a Hamiltonian. These problems may not fit naturally on the physical layout of the qubits. Our algorithms use a sequence of parameterized unitaries that sit on the qubit layout to produce quantum states depending on those parameters. Measurements of the objective function (or Hamiltonian) guide the choice of new parameters with the goal of moving the objective function up (or lowering the energy). As an example we consider finding approximate solutions to MaxCut on 3-regular graphs whereas the hardware is physical qubits laid out on a rectangular grid. We prove that the lowest depth version of the Quantum Approximate Optimization Algorithm will achieve an approximation ratio of at least 0.5293 on all large enough instances which beats random guessing (0.5). We open up the algorithm to have different parameters for each single qubit rotation and for each interaction associated with the nearest neighbor interactions on the grid. Small numerical experiments indicate that an enveloping classical algorithm can be used to find the parameters which sit on the grid to optimize an objective function with a …

CM Quintana, Yu Chen, Daniel Sank, AG Petukhov, TC White, Dvir Kafri, Ben Chiaro, Anthony Megrant, Rami Barends, Brooks Campbell, Zijun Chen, Andrew Dunsworth, Austin G Fowler, Rob Graff, Evan Jeffrey, Julian Kelly, Erik Lucero, JY Mutus, Matthew Neeley, Charles Neill, PJJ O’Malley, Pedram Roushan, Alireza Shabani, VN Smelyanskiy, Amit Vainsencher, James Wenner, Hartmut Neven, John M Martinis · Physical review letters 118 (5), 057702, 2017

By analyzing the dissipative dynamics of a tunable gap flux qubit, we extract both sides of its two-sided environmental flux noise spectral density over a range of frequencies around 2 k B T/h≈ 1 GHz, allowing for the observation of a classical-quantum crossover. Below the crossover point, the symmetric noise component follows a 1/f power law that matches the magnitude of the 1/f noise near 1 Hz. The antisymmetric component displays a 1/T dependence below 100 mK, providing dynamical evidence for a paramagnetic environment. Extrapolating the two-sided spectrum predicts the linewidth and reorganization energy of incoherent resonant tunneling between flux qubit wells.

Ryan Babbush, Dominic W Berry, Yuval R Sanders, Ian D Kivlichan, Artur Scherer, Annie Y Wei, Peter J Love, Alán Aspuru-Guzik · Quantum Science and Technology 3 (1), 015006, 2017

We present a quantum algorithm for the simulation of molecular systems that is asymptotically more efficient than all previous algorithms in the literature in terms of the main problem parameters. As in Babbush et al (2016 New Journal of Physics 18, 033032), we employ a recently developed technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The algorithm of this paper involves simulation under an oracle for the sparse, first-quantized representation of the molecular Hamiltonian known as the configuration interaction (CI) matrix. We construct and query the CI matrix oracle to allow for on-the-fly computation of molecular integrals in a way that is exponentially more efficient than classical numerical methods. Whereas second-quantized representations of the wavefunction require qubits, where N is the number of single-particle …

Ian D Kivlichan, Nathan Wiebe, Ryan Babbush, Alán Aspuru-Guzik · Journal of Physics A: Mathematical and Theoretical 50 (30), 305301, 2017

We present a quantum algorithm for simulating the dynamics of a first-quantized Hamiltonian in real space based on the truncated Taylor series algorithm. We avoid the possibility of singularities by applying various cutoffs to the system and using a high-order finite difference approximation to the kinetic energy operator. We find that our algorithm can simulate η interacting particles using a number of calculations of the pairwise interactions that scales, for a fixed spatial grid spacing, as , versus the time required by previous methods (assuming the number of orbitals is proportional to η), and scales super-polynomially better with the error tolerance than algorithms based on the Lie–Trotter–Suzuki product formula. Finally, we analyze discretization errors that arise from the spatial grid and show that under some circumstances these errors can remove the exponential speedups typically afforded by quantum …

Zhang Jiang, Vadim N Smelyanskiy, Sergei V Isakov, Sergio Boixo, Guglielmo Mazzola, Matthias Troyer, Hartmut Neven · Physical Review A 95 (1), 012322, 2017

We develop an instantonic calculus to derive an analytical expression for the thermally assisted tunneling decay rate of a metastable state in a fully connected quantum spin model. The tunneling decay problem can be mapped onto the Kramers escape problem of a classical random dynamical field. This dynamical field is simulated efficiently by path-integral quantum Monte Carlo (QMC). We show analytically that the exponential scaling with the number of spins of the thermally assisted quantum tunneling rate and the escape rate of the QMC process are identical. We relate this effect to the existence of a dominant instantonic tunneling path. The instanton trajectory is described by nonlinear dynamical mean-field theory equations for a single-site magnetization vector, which we solve exactly. Finally, we derive scaling relations for the “spiky” barrier shape when the spin tunneling and QMC rates scale polynomially with …

Craig Gidney · arXiv preprint arXiv:1706.07884, 2017

We present reversible classical circuits for performing various arithmetic operations aided by dirty ancillae (i.e. extra qubits in an unknown state that must be restored before the circuit ends). We improve the number of clean qubits needed to factor an n-bit number with Shor's algorithm from 1.5n+O(1) to n+2, assisted by n-1 dirty qubits, without increasing the asymptotic size or depth of the circuit.

Sergio Boixo, Vadim N Smelyanskiy, Hartmut Neven · arXiv preprint arXiv:1708.01875, 2017

Sampling from the output distribution of chaotic quantum evolutions, and of pseudo-random universal quantum circuits in particular, has been proposed as a prominent milestone for near-term quantum supremacy. The same paper notes that chaotic distributions are very sensitive to noise, and under quite general noise models converge to the uniform distribution over bit-strings exponentially in the number of gates. On the one hand, for increasing number of gates, it suffices to choose bit-strings at random to approximate the noisy distribution with fixed statistical distance. On the other hand, cross-entropy benchmarking can be used to gauge the fidelity of an experiment, and the distance to the uniform distribution. We estimate that state-of-the-art classical supercomputers would fail to simulate high-fidelity chaotic quantum circuits with approximately fifty qubits and depth forty. A recent interesting paper proposed a different approximation algorithm to a noisy distribution, extending previous results on the Fourier analysis of commuting quantum circuits. Using the statistical properties of the Porter-Thomas distribution, we show that this new approximation algorithm does not improve random guessing, in polynomial time. Therefore, it confirms previous results and does not represent an additional challenge to the suggested failure stated above.

Jarrod R McClean, Jonathan Romero, Ryan Babbush, Alán Aspuru-Guzik · New Journal of Physics 18 (2), 023023, 2016

Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as' the quantum variational eigensolver'was developed (Peruzzo et al 2014 Nat. Commun. 5 4213) with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through a relaxation of exponential operator splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this …

Peter JJ O’Malley, Ryan Babbush, Ian D Kivlichan, Jonathan Romero, Jarrod R McClean, Rami Barends, Julian Kelly, Pedram Roushan, Andrew Tranter, Nan Ding, Brooks Campbell, Yu Chen, Zijun Chen, Ben Chiaro, Andrew Dunsworth, Austin G Fowler, Evan Jeffrey, Erik Lucero, Anthony Megrant, Josh Y Mutus, Matthew Neeley, Charles Neill, Chris Quintana, Daniel Sank, Amit Vainsencher, Jim Wenner, Ted C White, Peter V Coveney, Peter J Love, Hartmut Neven, Alain Aspuru-Guzik, John M Martinis · Physical Review X 6 (3), 031007, 2016

We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable …

Vasil S Denchev, Sergio Boixo, Sergei V Isakov, Nan Ding, Ryan Babbush, Vadim Smelyanskiy, John Martinis, Hartmut Neven · Physical Review X 6 (3), 031015, 2016

Quantum annealing (QA) has been proposed as a quantum enhanced optimization heuristic exploiting tunneling. Here, we demonstrate how finite-range tunneling can provide considerable computational advantage. For a crafted problem designed to have tall and narrow energy barriers separating local minima, the D-Wave 2X quantum annealer achieves significant runtime advantages relative to simulated annealing (SA). For instances with 945 variables, this results in a time-to-99%-success-probability that is∼ 1 0 8 times faster than SA running on a single processor core. We also compare physical QA with the quantum Monte Carlo algorithm, an algorithm that emulates quantum tunneling on classical processors. We observe a substantial constant overhead against physical QA: D-Wave 2X again runs up to∼ 1 0 8 times faster than an optimized implementation of the quantum Monte Carlo algorithm on a single …

Rami Barends, Alireza Shabani, Lucas Lamata, Julian Kelly, Antonio Mezzacapo, U Las Heras, Ryan Babbush, Austin G Fowler, Brooks Campbell, Yu Chen, Zijun Chen, Ben Chiaro, Andrew Dunsworth, Evan Jeffrey, Erik Lucero, Anthony Megrant, JY Mutus, Matthew Neeley, Charles Neill, PJJ O’Malley, Chris Quintana, Pedran Roushan, D Sank, Amit Vainsencher, James Wenner, TC White, Enrique Solano, Hartmut Neven, John M Martinis · Nature 534 (7606), 222-226, 2016

Quantum mechanics can help to solve complex problems in physics and chemistry, provided they can be programmed in a physical device. In adiabatic quantum computing,,, a system is slowly evolved from the ground state of a simple initial Hamiltonian to a final Hamiltonian that encodes a computational problem. The appeal of this approach lies in the combination of simplicity and generality; in principle, any problem can be encoded. In practice, applications are restricted by limited connectivity, available interactions and noise. A complementary approach is digital quantum computing, which enables the construction of arbitrary interactions and is compatible with error correction,, but uses quantum circuit algorithms that are problem-specific. Here we combine the advantages of both approaches by implementing digitized adiabatic quantum computing in a superconducting system. We tomographically probe the …

Charles Neill, P Roushan, M Fang, Y Chen, M Kolodrubetz, Z Chen, A Megrant, R Barends, B Campbell, B Chiaro, A Dunsworth, E Jeffrey, J Kelly, J Mutus, PJJ O’malley, C Quintana, D Sank, A Vainsencher, J Wenner, TC White, Anatoli Polkovnikov, JM Martinis · Nature Physics 12 (11), 1037-1041, 2016

Statistical mechanics is founded on the assumption that all accessible configurations of a system are equally likely. This requires dynamics that explore all states over time, known as ergodic dynamics. In isolated quantum systems, however, the occurrence of ergodic behaviour has remained an outstanding question,,,. Here, we demonstrate ergodic dynamics in a small quantum system consisting of only three superconducting qubits. The qubits undergo a sequence of rotations and interactions and we measure the evolution of the density matrix. Maps of the entanglement entropy show that the full system can act like a reservoir for individual qubits, increasing their entropy through entanglement. Surprisingly, these maps bear a strong resemblance to the phase space dynamics in the classical limit; classically, chaotic motion coincides with higher entanglement entropy. We further show that in regions of high entropy …

Sergio Boixo, Vadim N Smelyanskiy, Alireza Shabani, Sergei V Isakov, Mark Dykman, Vasil S Denchev, Mohammad H Amin, Anatoly Yu Smirnov, Masoud Mohseni, Hartmut Neven · Nature communications 7 (1), 10327, 2016

Quantum tunnelling is a phenomenon in which a quantum state traverses energy barriers higher than the energy of the state itself. Quantum tunnelling has been hypothesized as an advantageous physical resource for optimization in quantum annealing. However, computational multiqubit tunnelling has not yet been observed, and a theory of co-tunnelling under high- and low-frequency noises is lacking. Here we show that 8-qubit tunnelling plays a computational role in a currently available programmable quantum annealer. We devise a probe for tunnelling, a computational primitive where classical paths are trapped in a false minimum. In support of the design of quantum annealers we develop a nonperturbative theory of open quantum dynamics under realistic noise characteristics. This theory accurately predicts the rate of many-body dissipative quantum tunnelling subject to the polaron effect. Furthermore, we …

Ryan Babbush, Dominic W Berry, Ian D Kivlichan, Annie Y Wei, Peter J Love, Alán Aspuru-Guzik · New Journal of Physics 18 (3), 033032, 2016

We introduce novel algorithms for the quantum simulation of fermionic systems which are dramatically more efficient than those based on the Lie–Trotter–Suzuki decomposition. We present the first application of a general technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The key difficulty in applying algorithms for general sparse Hamiltonian simulation to fermionic simulation is that a query, corresponding to computation of an entry of the Hamiltonian, is costly to compute. This means that the gate complexity would be much higher than quantified by the query complexity. We solve this problem with a novel quantum algorithm for on-the-fly computation of integrals that is exponentially faster than classical sampling. While the approaches presented here are readily applicable to a wide class of fermionic models, we focus on …

Sergei V Isakov, Guglielmo Mazzola, Vadim N Smelyanskiy, Zhang Jiang, Sergio Boixo, Hartmut Neven, Matthias Troyer · Physical review letters 117 (18), 180402, 2016

The tunneling between the two ground states of an Ising ferromagnet is a typical example of many-body tunneling processes between two local minima, as they occur during quantum annealing. Performing quantum Monte Carlo (QMC) simulations we find that the QMC tunneling rate displays the same scaling with system size, as the rate of incoherent tunneling. The scaling in both cases is O (Δ 2), where Δ is the tunneling splitting (or equivalently the minimum spectral gap). An important consequence is that QMC simulations can be used to predict the performance of a quantum annealer for tunneling through a barrier. Furthermore, by using open instead of periodic boundary conditions in imaginary time, equivalent to a projector QMC algorithm, we obtain a quadratic speedup for QMC simulations, and achieve linear scaling in Δ. We provide a physical understanding of these results and their range of applicability …

Jirawat Tangpanitanon, Victor M Bastidas, Sarah Al-Assam, Pedram Roushan, Dieter Jaksch, Dimitris G Angelakis · Physical Review Letters 117 (21), 213603, 2016

We show how to implement topological or Thouless pumping of interacting photons in one-dimensional nonlinear resonator arrays by simply modulating the frequency of the resonators periodically in space and time. The interplay between the interactions and the adiabatic modulations enables robust transport of Fock states with few photons per site. We analyze the transport mechanism via an effective analytic model and study its topological properties and its protection to noise. We conclude by a detailed study of an implementation with existing circuit-QED architectures.

Theodore C White, JY Mutus, Justin Dressel, J Kelly, R Barends, E Jeffrey, D Sank, A Megrant, B Campbell, Yu Chen, Z Chen, B Chiaro, A Dunsworth, IC Hoi, C Neill, PJJ O’malley, P Roushan, A Vainsencher, J Wenner, AN Korotkov, John M Martinis · npj Quantum Information 2 (1), 1-5, 2016

Weak measurement has provided new insight into the nature of quantum measurement, by demonstrating the ability to extract average state information without fully projecting the system. For single-qubit measurements, this partial projection has been demonstrated with violations of the Leggett–Garg inequality. Here we investigate the effects of weak measurement on a maximally entangled Bell state through application of the Hybrid Bell–Leggett–Garg inequality (BLGI) on a linear chain of four transmon qubits. By correlating the results of weak ancilla measurements with subsequent projective readout, we achieve a violation of the BLGI with 27 sds of certainty.

Marek M Rams, Masoud Mohseni, Adolfo Del Campo · New Journal of Physics 18 (12), 123034, 2016

We introduce an inhomogeneous protocol to drive a weakly disordered quantum spin chain quasi-adiabatically across a quantum phase transition and minimize the residual energy of the final state. The number of spins that simultaneously reach the critical point is controlled by the length scale in which the magnetic field is modulated, introducing an effective size that favors adiabatic dynamics. The dependence of the residual energy on this length scale and the velocity at which the magnetic field sweeps out the chain is shown to be nonmonotonic. We determine the conditions for an optimal suppression of the residual energy of the final state and show that inhomogeneous driving can outperform conventional adiabatic schemes based on homogeneous control fields by several orders of magnitude.

Julian Kelly, Rami Barends, Austin G Fowler, Anthony Megrant, Evan Jeffrey, Theodore C White, Daniel Sank, Josh Y Mutus, Brooks Campbell, Yu Chen, Zijun Chen, Ben Chiaro, Andrew Dunsworth, I-C Hoi, Charles Neill, Peter JJ O’Malley, Christopher Quintana, Pedran Roushan, Amit Vainsencher, James Wenner, Andrew N Cleland, John M Martinis · Nature 519 (7541), 66-69, 2015

Quantum computing becomes viable when a quantum state can be protected from environment-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC),,,,, is capable of identifying and correcting them. Adding more qubits improves the preservation of states by guaranteeing that increasingly larger clusters of errors will not cause logical failure—a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural step towards the two-dimensional surface code QEC scheme, and track errors as they occur by repeatedly performing projective quantum non …

Rami Barends, Lucas Lamata, Julian Kelly, L García-Álvarez, Austin G Fowler, A Megrant, Evan Jeffrey, Ted C White, Daniel Sank, Josh Y Mutus, B Campbell, Yu Chen, Zhaoshi Chen, B Chiaro, A Dunsworth, I-C Hoi, C Neill, PJJ O’Malley, Céline Quintana, P Roushan, A Vainsencher, J Wenner, E Solano, John M Martinis · Nature communications 6 (1), 7654, 2015

One of the key applications of quantum information is simulating nature. Fermions are ubiquitous in nature, appearing in condensed matter systems, chemistry and high energy physics. However, universally simulating their interactions is arguably one of the largest challenges, because of the difficulties arising from anticommutativity. Here we use digital methods to construct the required arbitrary interactions, and perform quantum simulation of up to four fermionic modes with a superconducting quantum circuit. We employ in excess of 300 quantum logic gates, and reach fidelities that are consistent with a simple model of uncorrelated errors. The presented approach is in principle scalable to a larger number of modes, and arbitrary spatial dimensions.

Ya Wang, Florian Dolde, Jacob Biamonte, Ryan Babbush, Ville Bergholm, Sen Yang, Ingmar Jakobi, Philipp Neumann, Alán Aspuru-Guzik, James D Whitfield, Jorg Wrachtrup · ACS nano 9 (8), 7769-7774, 2015

Ab initio computation of molecular properties is one of the most promising applications of quantum computing. While this problem is widely believed to be intractable for classical computers, efficient quantum algorithms exist which have the potential to vastly accelerate research throughput in fields ranging from material science to drug discovery. Using a solid-state quantum register realized in a nitrogen-vacancy (NV) defect in diamond, we compute the bond dissociation curve of the minimal basis helium hydride cation, HeH+. Moreover, we report an energy uncertainty (given our model basis) of the order of 10–14 hartree, which is 10 orders of magnitude below the desired chemical precision. As NV centers in diamond provide a robust and straightforward platform for quantum information processing, our work provides an important step toward a fully scalable solid-state implementation of a quantum chemistry …

Xiaoting Wang, Michele Allegra, Kurt Jacobs, Seth Lloyd, Cosmo Lupo, Masoud Mohseni · Physical review letters 114 (17), 170501, 2015

Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for a wide range of quantum control problems. So far, this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here we show that the quantum brachistochrone problem can be recast as that of finding geodesic paths in the space of unitary operators. We expect this brachistochrone-geodesic connection to have broad applications, as it opens up minimal-time control to the tools of geometry. As one such application, we use it to obtain a fast numerical method to solve the brachistochrone problem, and apply this method to two examples demonstrating its power.

Michael R Geller, Emmanuel Donate, Yu Chen, Michael T Fang, Nelson Leung, Charles Neill, Pedram Roushan, John M Martinis · Physical Review A 92 (1), 012320, 2015

We study a recently demonstrated design for a high-performance tunable coupler suitable for superconducting Xmon and planar transmon qubits [Y. Chen et al., Phys. Rev. Lett. 113, 220502 (2014)]. The coupler circuit uses a single flux-biased Josephson junction and acts as a tunable current divider. We calculate the effective qubit-qubit interaction Hamiltonian by treating the nonlinearity of the qubit and coupler junctions perturbatively. We find that the qubit nonlinearity has two principal effects: The first is to suppress the magnitude of the transverse σ x⊗ σ x coupling from that obtained in the harmonic approximation by about 15%, assuming typical qubit parameters. The second is to induce a small diagonal σ z⊗ σ z coupling. The effects of the coupler junction nonlinearity are negligible in the parameter regime considered. The approach used here can be applied to other complex nonlinear circuits arising in the …

Sergio Boixo, Gerardo Ortiz, Rolando Somma · The European Physical Journal Special Topics 224 (1), 35-49, 2015

Discrete combinatorial optimization consists in finding the optimal configuration that minimizes a given discrete objective function. An interpretation of such a function as the energy of a classical system allows us to reduce the optimization problem into the preparation of a low-temperature thermal state of the system. Motivated by the quantum annealing method, we present three strategies to prepare the low-temperature state that exploit quantum mechanics in remarkable ways. We focus on implementations without uncontrolled errors induced by the environment. This allows us to rigorously prove a quantum advantage. The first strategy uses a classical-to-quantum mapping, where the equilibrium properties of a classical system in d spatial dimensions can be determined from the ground state properties of a quantum system also in d spatial dimensions. We show how such a ground state can be prepared by …

Troels F Rønnow, Zhihui Wang, Joshua Job, Sergio Boixo, Sergei V Isakov, David Wecker, John M Martinis, Daniel A Lidar, Matthias Troyer · science 345 (6195), 420-424, 2014

The development of small-scale quantum devices raises the question of how to fairly assess and detect quantum speedup. Here, we show how to define and measure quantum speedup and how to avoid pitfalls that might mask or fake such a speedup. We illustrate our discussion with data from tests run on a D-Wave Two device with up to 503 qubits. By using random spin glass instances as a benchmark, we found no evidence of quantum speedup when the entire data set is considered and obtained inconclusive results when comparing subsets of instances on an instance-by-instance basis. Our results do not rule out the possibility of speedup for other classes of problems and illustrate the subtle nature of the quantum speedup question.

Yu Chen, C Neill, Pedram Roushan, Nelson Leung, Michael Fang, Rami Barends, Julian Kelly, Brooks Campbell, Z Chen, Benjamin Chiaro, Andrew Dunsworth, Evan Jeffrey, Anthony Megrant, JY Mutus, PJJ O’Malley, CM Quintana, D Sank, Amit Vainsencher, J Wenner, TC White, Michael R Geller, Andrew N Cleland, John M Martinis · Physical review letters 113 (22), 220502, 2014

We introduce a superconducting qubit architecture that combines high-coherence qubits and tunable qubit-qubit coupling. With the ability to set the coupling to zero, we demonstrate that this architecture is protected from the frequency crowding problems that arise from fixed coupling. More importantly, the coupling can be tuned dynamically with nanosecond resolution, making this architecture a versatile platform with applications ranging from quantum logic gates to quantum simulation. We illustrate the advantages of dynamical coupling by implementing a novel adiabatic controlled-z gate, with a speed approaching that of single-qubit gates. Integrating coherence and scalable control, the introduced qubit architecture provides a promising path towards large-scale quantum computation and simulation.

Trevor Lanting, Anthony J Przybysz, A Yu Smirnov, Federico M Spedalieri, Mohammad H Amin, Andrew J Berkley, Richard Harris, Fabio Altomare, Sergio Boixo, Paul Bunyk, Neil Dickson, C Enderud, Jeremy P Hilton, Emile Hoskinson, Mark W Johnson, Eric Ladizinsky, Nicholas Ladizinsky, Richard Neufeld, Travis Oh, Ilya Perminov, Chris Rich, Murray C Thom, E Tolkacheva, Sergey Uchaikin, AB Wilson, Geordie Rose · Physical Review X 4 (2), 021041, 2014

Entanglement lies at the core of quantum algorithms designed to solve problems that are intractable by classical approaches. One such algorithm, quantum annealing (QA), provides a promising path to a practical quantum processor. We have built a series of architecturally scalable QA processors consisting of networks of manufactured interacting spins (qubits). Here, we use qubit tunneling spectroscopy to measure the energy eigenspectrum of two-and eight-qubit systems within one such processor, demonstrating quantum coherence in these systems. We present experimental evidence that, during a critical portion of QA, the qubits become entangled and entanglement persists even as these systems reach equilibrium with a thermal environment. Our results provide an encouraging sign that QA is a viable technology for large-scale quantum computing.

Nan Ding, Youhan Fang, Ryan Babbush, Changyou Chen, Robert D Skeel, Hartmut Neven · Advances in neural information processing systems 27, 2014

Dynamics-based sampling methods, such as Hybrid Monte Carlo (HMC) and Langevin dynamics (LD), are commonly used to sample target distributions. Recently, such approaches have been combined with stochastic gradient techniques to increase sampling efficiency when dealing with large datasets. An outstanding problem with this approach is that the stochastic gradient introduces an unknown amount of noise which can prevent proper sampling after discretization. To remedy this problem, we show that one can leverage a small number of additional variables in order to stabilize momentum fluctuations induced by the unknown noise. Our method is inspired by the idea of a thermostat in statistical physics and is justified by a general theory.

Pedram Roushan, C Neill, Yu Chen, M Kolodrubetz, C Quintana, N Leung, M Fang, R Barends, B Campbell, Z Chen, B Chiaro, A Dunsworth, E Jeffrey, J Kelly, A Megrant, J Mutus, PJJ O’Malley, D Sank, A Vainsencher, J Wenner, T White, A Polkovnikov, AN Cleland, JM Martinis · Nature 515 (7526), 241-244, 2014

Topology, with its abstract mathematical constructs, often manifests itself in physics and has a pivotal role in our understanding of natural phenomena. Notably, the discovery of topological phases in condensed-matter systems has changed the modern conception of phases of matter,,,,. The global nature of topological ordering, however, makes direct experimental probing an outstanding challenge. Present experimental tools are mainly indirect and, as a result, are inadequate for studying the topology of physical systems at a fundamental level. Here we employ the exquisite control afforded by state-of-the-art superconducting quantum circuits to investigate topological properties of various quantum systems. The essence of our approach is to infer geometric curvature by measuring the deflection of quantum trajectories in the curved space of the Hamiltonian. Topological properties are then revealed by integrating the …

Sergio Boixo, Vadim N Smelyanskiy, Alireza Shabani, Sergei V Isakov, Mark Dykman, Vasil S Denchev, Mohammad Amin, Anatoly Smirnov, Masoud Mohseni, Hartmut Neven · arXiv preprint arXiv:1411.4036, 2014

Quantum tunneling is a phenomenon in which a quantum state traverses energy barriers above the energy of the state itself. Tunneling has been hypothesized as an advantageous physical resource for optimization. Here we present the first experimental evidence of a computational role of multiqubit quantum tunneling in the evolution of a programmable quantum annealer. We develop a theoretical model based on a NIBA Quantum Master Equation to describe the multiqubit dissipative tunneling effects under the complex noise characteristics of such quantum devices. We start by considering a computational primitive, an optimization problem consisting of just one global and one false minimum. The quantum evolutions enable tunneling to the global minimum while the corresponding classical paths are trapped in a false minimum. In our study the non-convex potentials are realized by frustrated networks of qubit clusters with strong intra-cluster coupling. We show that the collective effect of the quantum environment is suppressed in the "critical" phase during the evolution where quantum tunneling "decides" the right path to solution. In a later stage dissipation facilitates the multiqubit tunneling leading to the solution state. The predictions of the model accurately describe the experimental data from the D-Wave Two quantum annealer at NASA Ames. In our computational primitive the temperature dependence of the probability of success in the quantum model is opposite to that of the classical paths with thermal hopping. Specifically, we provide an analysis of an optimization problem with sixteen qubits, demonstrating eight qubit tunneling that …

Seth Lloyd, Masoud Mohseni, Patrick Rebentrost · arXiv preprint arXiv:1307.0411, 2013

Machine-learning tasks frequently involve problems of manipulating and classifying large numbers of vectors in high-dimensional spaces. Classical algorithms for solving such problems typically take time polynomial in the number of vectors and the dimension of the space. Quantum computers are good at manipulating high-dimensional vectors in large tensor product spaces. This paper provides supervised and unsupervised quantum machine learning algorithms for cluster assignment and cluster finding. Quantum machine learning can take time logarithmic in both the number of vectors and their dimension, an exponential speed-up over classical algorithms.

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