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The quantum simulation of electronic structure is one of the most promising applications of quantum computers. It has potential applications to materials and drug design. This tutorial provides an introduction to OpenFermion, a library for obtaining and manipulating representations of fermionic and qubit Hamiltonians as well as compiling quantum simulation circuits in Cirq.
try:
import openfermion as of
import openfermionpyscf as ofpyscf
except ImportError:
print("Installing OpenFermion and OpenFermion-PySCF...")
!pip install openfermion openfermionpyscf --quiet
import numpy as np
from scipy.sparse import linalg
import cirq
import openfermion as of
import openfermionpyscf as ofpyscf
Background
A system of \(N\) fermionic modes is described by a set of fermionic annihilation operators \(\{a_p\}_{p=0}^{N-1}\) satisfying the canonical anticommutation relations \(\begin{aligned} \{a_p, a_q\} &= 0, \\ \{a_p, a^\dagger_q\} &= \delta_{pq}, \end{aligned}\) where \(\{A, B\} := AB + BA\). The adjoint \(a^\dagger_p\) of an annihilation operator \(a_p\) is called a creation operator, and we refer to creation and annihilation operators as fermionic ladder operators.
The canonical anticommutation relations impose a number of consequences on the structure of the vector space on which the ladder operators act; see Michael Nielsen's notes for a good discussion.
The electronic structure Hamiltonian is commonly written in the form
\[ \sum_{pq} T_{pq} a_p^\dagger a_q + \sum_{pqrs} V_{pqrs} a_p^\dagger a_q^\dagger a_r a_s \]
where the \(T_{pq}\) and \(V_{pqrs}\) are coefficients which depend on the physical system being described. We are interested in calculating the lowest eigenvalue of the Hamiltonian. This eigenvalue is also called the ground state energy.
FermionOperator and QubitOperator
openfermion.FermionOperator
- Stores a weighted sum (linear combination) of fermionic terms
- A fermionic term is a product of ladder operators
- Examples of things that can be represented by
FermionOperator:
\[ \begin{align} & a_1 \nonumber \\ & 1.7 a^\dagger_3 \nonumber \\ &-1.7 \, a^\dagger_3 a_1 \nonumber \\ &(1 + 2i) \, a^\dagger_4 a^\dagger_3 a_9 a_1 \nonumber \\ &(1 + 2i) \, a^\dagger_4 a^\dagger_3 a_9 a_1 - 1.7 \, a^\dagger_3 a_1 \nonumber \end{align} \]
- A fermionic term is internally represented as a tuple of tuples
- Each inner tuple represents a single ladder operator as (index, action)
- Examples of fermionic terms:
\[ \begin{align} I & \mapsto () \nonumber \\ a_1 & \mapsto ((1, 0),) \nonumber \\ a^\dagger_3 & \mapsto ((3, 1),) \nonumber \\ a^\dagger_3 a_1 & \mapsto ((3, 1), (1, 0)) \nonumber \\ a^\dagger_4 a^\dagger_3 a_9 a_1 & \mapsto ((4, 1), (3, 1), (9, 0), (1, 0)) \nonumber \end{align} \]
FermionOperatoris a sum of terms, represented as a dictionary from term to coefficient
op = of.FermionOperator(((4, 1), (3, 1), (9, 0), (1, 0)), 1 + 2j) + of.FermionOperator(
((3, 1), (1, 0)), -1.7
)
print(op.terms)
{((4, 1), (3, 1), (9, 0), (1, 0)): (1+2j), ((3, 1), (1, 0)): -1.7}
Alternative notation, useful when playing around:
\[ \begin{align} I & \mapsto \textrm{""} \nonumber \\ a_1 & \mapsto \textrm{"1"} \nonumber \\ a^\dagger_3 & \mapsto \textrm{"3^"} \nonumber \\ a^\dagger_3 a_1 & \mapsto \textrm{"3^}\;\textrm{1"} \nonumber \\ a^\dagger_4 a^\dagger_3 a_9 a_1 & \mapsto \textrm{"4^}\;\textrm{3^}\;\textrm{9}\;\textrm{1"} \nonumber \end{align} \]
op = of.FermionOperator("4^ 3^ 9 1", 1 + 2j) + of.FermionOperator("3^ 1", -1.7)
print(op.terms)
{((4, 1), (3, 1), (9, 0), (1, 0)): (1+2j), ((3, 1), (1, 0)): -1.7}
Just print the operator for a nice readable representation:
print(op)
-1.7 [3^ 1] + (1+2j) [4^ 3^ 9 1]
openfermion.QubitOperator
Same as FermionOperator, but the possible actions are 'X', 'Y', and 'Z' instead of 1 and 0.
op = of.QubitOperator(((1, "X"), (2, "Y"), (3, "Z")))
op += of.QubitOperator("X3 Z4", 3.0)
print(op)
1.0 [X1 Y2 Z3] + 3.0 [X3 Z4]
FermionOperator and QubitOperator actually inherit from the same parent class: openfermion.SymbolicOperator.
The Jordan-Wigner and Bravyi-Kitaev transforms
A fermionic transform maps FermionOperators to QubitOperators in a way that preserves the canonical anticommutation relations. The most basic transforms are the Jordan-Wigner transform (JWT) and Bravyi-Kitaev transform (BKT). Note that the BKT requires the total number of qubits to be predetermined. Whenever a fermionic transform is being applied implicitly, it is the JWT.
op = of.FermionOperator("2^ 15")
print(of.jordan_wigner(op))
print()
print(of.bravyi_kitaev(op, n_qubits=16))
(0.25+0j) [X2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z13 Z14 X15] + 0.25j [X2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z13 Z14 Y15] + -0.25j [Y2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z13 Z14 X15] + (0.25+0j) [Y2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z13 Z14 Y15] (-0.25+0j) [Z1 X2 X3 X7 Z15] + -0.25j [Z1 X2 X3 Y7 Z11 Z13 Z14] + 0.25j [Z1 Y2 X3 X7 Z15] + (-0.25+0j) [Z1 Y2 X3 Y7 Z11 Z13 Z14]
Exercise
Below are some examples of how FermionOperators are mapped to QubitOperators by the Jordan-Wigner transform (the notation 'h.c.' stands for 'hermitian conjugate'):
\[ \begin{align*} a_p^\dagger &\mapsto \frac12 (X_p - i Y_p) Z_0 \cdots Z_{p-1}\\ a_p^\dagger a_p &\mapsto \frac12 (I - Z_p)\\ (\beta a_p^\dagger a_q + \text{h.c.}) &\mapsto \frac12 [\text{Re}(\beta) (X_p ZZ \cdots ZZ X_q + Y_p ZZ \cdots ZZ Y_q) + \text{Im}(\beta) (Y_p ZZ \cdots ZZ X_q - X_p ZZ \cdots ZZ Y_q)] \end{align*} \]
Verify these mappings for \(p=2\) and \(q=7\). The openfermion.hermitian_conjugated function may be useful here.
a2 = of.FermionOperator("2")
print(of.jordan_wigner(a2))
print()
a2dag = of.FermionOperator("2^")
print(of.jordan_wigner(a2dag * a2))
print()
a7 = of.FermionOperator("7")
a7dag = of.FermionOperator("7^")
print(of.jordan_wigner((1 + 2j) * (a2dag * a7) + (1 - 2j) * (a7dag * a2)))
0.5 [Z0 Z1 X2] + 0.5j [Z0 Z1 Y2] (0.5+0j) [] + (-0.5+0j) [Z2] (0.5+0j) [X2 Z3 Z4 Z5 Z6 X7] + (-1+0j) [X2 Z3 Z4 Z5 Z6 Y7] + (1+0j) [Y2 Z3 Z4 Z5 Z6 X7] + (0.5+0j) [Y2 Z3 Z4 Z5 Z6 Y7]
Solution
a2 = of.FermionOperator("2")
a2dag = of.FermionOperator("2^")
a7 = of.FermionOperator("7")
a7dag = of.FermionOperator("7^")
print(of.jordan_wigner(a2dag))
print()
print(of.jordan_wigner(a2dag * a2))
print()
op = (2 + 3j) * a2dag * a7
op += of.hermitian_conjugated(op)
print(of.jordan_wigner(op))
0.5 [Z0 Z1 X2] + -0.5j [Z0 Z1 Y2] (0.5+0j) [] + (-0.5+0j) [Z2] (1+0j) [X2 Z3 Z4 Z5 Z6 X7] + (-1.5+0j) [X2 Z3 Z4 Z5 Z6 Y7] + (1.5+0j) [Y2 Z3 Z4 Z5 Z6 X7] + (1+0j) [Y2 Z3 Z4 Z5 Z6 Y7]
Exercise
Use the + and * operators to verify that after applying the JWT to ladder operators, the resulting QubitOperators satisfy
\[ \begin{align} a_2 a_7 + a_7 a_2 &= 0 \\ a_2 a_7^\dagger + a_7^\dagger a_2 &= 0\\ a_2 a_2^\dagger + a_2^\dagger a_2 &= 1 \end{align} \]
Solution
a2_jw = of.jordan_wigner(a2)
a2dag_jw = of.jordan_wigner(a2dag)
a7_jw = of.jordan_wigner(a7)
a7dag_jw = of.jordan_wigner(a7dag)
print(a2_jw * a7_jw + a7_jw * a2_jw)
print(a2_jw * a7dag_jw + a7dag_jw * a2_jw)
print(a2_jw * a2dag_jw + a2dag_jw * a2_jw)
0 0 (1+0j) []
Array data structures
- When
FermionOperators have specialized structure we can store coefficients in numpy arrays, enabling fast numerical manipulation. - Array data structures can always be converted to
FermionOperatorusingopenfermion.get_fermion_operator.
InteractionOperator
- Stores the one- and two-body tensors \(T_{pq}\) and \(V_{pqrs}\) of the molecular Hamiltonian
\[ \sum_{pq} T_{pq} a_p^\dagger a_q + \sum_{pqrs} V_{pqrs} a_p^\dagger a_q^\dagger a_r a_s \]
- Default data structure for molecular Hamiltonians
- Convert from
FermionOperatorusingopenfermion.get_interaction_operator
DiagonalCoulombHamiltonian
- Stores the one- and two-body coefficient matrices \(T_{pq}\) and \(V_{pq}\) of a Hamiltonian with a diagonal Coulomb term:
\[ \sum_{pq} T_{pq} a_p^\dagger a_q + \sum_{pq} V_{pq} a_p^\dagger a_p a_q^\dagger a_q \]
- Leads to especially efficient algorithms for quantum simulation
- Convert from
FermionOperatorusingopenfermion.get_diagonal_coulomb_hamiltonian
QuadraticHamiltonian
- Stores the Hermitian matrix \(M_{pq}\) and antisymmetric matrix \(\Delta_{pq}\) describing a general quadratic Hamiltonian
\[ \sum_{p, q} M_{pq} a^\dagger_p a_q + \frac12 \sum_{p, q} (\Delta_{pq} a^\dagger_p a^\dagger_q + \text{h.c.}) \]
- Routines included for efficient diagonalization (can handle thousands of fermionic modes)
- Convert from
FermionOperatorusingopenfermion.get_quadratic_hamiltonian
Generating the Hamiltonian for a molecule
The cell below demonstrates using one of our electronic structure package plugins, OpenFermion-PySCF, to generate a molecular Hamiltonian for a hydrogen molecule. Note that the Hamiltonian is returned as an InteractionOperator. We'll convert it to a FermionOperator and print the result.
# Set molecule parameters
geometry = [("H", (0.0, 0.0, 0.0)), ("H", (0.0, 0.0, 0.8))]
basis = "sto-3g"
multiplicity = 1
charge = 0
# Perform electronic structure calculations and
# obtain Hamiltonian as an InteractionOperator
hamiltonian = ofpyscf.generate_molecular_hamiltonian(
geometry, basis, multiplicity, charge
)
# Convert to a FermionOperator
hamiltonian_ferm_op = of.get_fermion_operator(hamiltonian)
print(hamiltonian_ferm_op)
0.66147151365 [] + -1.2178260299951058 [0^ 0] + 0.3316650744318082 [0^ 0^ 0 0] + 0.09231339177803066 [0^ 0^ 2 2] + 0.3316650744318082 [0^ 1^ 1 0] + 0.09231339177803066 [0^ 1^ 3 2] + 0.09231339177803066 [0^ 2^ 0 2] + 0.3267206861819477 [0^ 2^ 2 0] + 0.09231339177803066 [0^ 3^ 1 2] + 0.3267206861819477 [0^ 3^ 3 0] + 0.3316650744318082 [1^ 0^ 0 1] + 0.09231339177803066 [1^ 0^ 2 3] + -1.2178260299951058 [1^ 1] + 0.3316650744318082 [1^ 1^ 1 1] + 0.09231339177803066 [1^ 1^ 3 3] + 0.09231339177803066 [1^ 2^ 0 3] + 0.3267206861819477 [1^ 2^ 2 1] + 0.09231339177803066 [1^ 3^ 1 3] + 0.3267206861819477 [1^ 3^ 3 1] + 0.32672068618194783 [2^ 0^ 0 2] + 0.09231339177803066 [2^ 0^ 2 0] + 0.32672068618194783 [2^ 1^ 1 2] + 0.09231339177803066 [2^ 1^ 3 0] + -0.5096378744364826 [2^ 2] + 0.09231339177803066 [2^ 2^ 0 0] + 0.34339576784573445 [2^ 2^ 2 2] + 0.09231339177803066 [2^ 3^ 1 0] + 0.34339576784573445 [2^ 3^ 3 2] + 0.32672068618194783 [3^ 0^ 0 3] + 0.09231339177803066 [3^ 0^ 2 1] + 0.32672068618194783 [3^ 1^ 1 3] + 0.09231339177803066 [3^ 1^ 3 1] + 0.09231339177803066 [3^ 2^ 0 1] + 0.34339576784573445 [3^ 2^ 2 3] + -0.5096378744364826 [3^ 3] + 0.09231339177803066 [3^ 3^ 1 1] + 0.34339576784573445 [3^ 3^ 3 3]
Let's calculate the ground energy (lowest eigenvalue) of the Hamiltonian. First, we'll map the FermionOperator to a QubitOperator using the JWT. Then, we'll convert the QubitOperator to a SciPy sparse matrix and get its lowest eigenvalue.
# Map to QubitOperator using the JWT
hamiltonian_jw = of.jordan_wigner(hamiltonian_ferm_op)
# Convert to Scipy sparse matrix
hamiltonian_jw_sparse = of.get_sparse_operator(hamiltonian_jw)
# Compute ground energy
eigs, _ = linalg.eigsh(hamiltonian_jw_sparse, k=1, which="SA")
ground_energy = eigs[0]
print("Ground_energy: {}".format(ground_energy))
print("JWT transformed Hamiltonian:")
print(hamiltonian_jw)
Ground_energy: -1.134147666677095 JWT transformed Hamiltonian: (-0.16733398905695201+0j) [] + (-0.04615669588901533+0j) [X0 X1 Y2 Y3] + (0.04615669588901533+0j) [X0 Y1 Y2 X3] + (0.04615669588901533+0j) [Y0 X1 X2 Y3] + (-0.04615669588901533+0j) [Y0 Y1 X2 X3] + (0.16251648748871642+0j) [Z0] + (0.1658325372159041+0j) [Z0 Z1] + (0.11720364720195856+0j) [Z0 Z2] + (0.1633603430909739+0j) [Z0 Z3] + (0.16251648748871636+0j) [Z1] + (0.1633603430909739+0j) [Z1 Z2] + (0.11720364720195856+0j) [Z1 Z3] + (-0.1974429369975584+0j) [Z2] + (0.17169788392286722+0j) [Z2 Z3] + (-0.19744293699755838+0j) [Z3]
Exercise
Compute the ground energy of the same Hamiltonian, but via the Bravyi-Kitaev transform. Verify that you get the same value.
# Map to QubitOperator using the JWT
hamiltonian_bk = of.bravyi_kitaev(hamiltonian_ferm_op)
# Convert to Scipy sparse matrix
hamiltonian_bk_sparse = of.get_sparse_operator(hamiltonian_bk)
# Compute ground energy
eigs, _ = linalg.eigsh(hamiltonian_bk_sparse, k=1, which="SA")
ground_energy = eigs[0]
print("Ground_energy: {}".format(ground_energy))
print("BK transformed Hamiltonian:")
print(hamiltonian_bk)
Ground_energy: -1.1341476666770918 BK transformed Hamiltonian: (-0.16733398905695201+0j) [] + (0.04615669588901533+0j) [X0 Z1 X2] + (0.04615669588901533+0j) [X0 Z1 X2 Z3] + (0.04615669588901533+0j) [Y0 Z1 Y2] + (0.04615669588901533+0j) [Y0 Z1 Y2 Z3] + (0.16251648748871642+0j) [Z0] + (0.16251648748871636+0j) [Z0 Z1] + (0.1633603430909739+0j) [Z0 Z1 Z2] + (0.1633603430909739+0j) [Z0 Z1 Z2 Z3] + (0.11720364720195856+0j) [Z0 Z2] + (0.11720364720195856+0j) [Z0 Z2 Z3] + (0.1658325372159041+0j) [Z1] + (-0.19744293699755838+0j) [Z1 Z2 Z3] + (0.17169788392286722+0j) [Z1 Z3] + (-0.1974429369975584+0j) [Z2]
Solution
# Map to QubitOperator using the BKT
hamiltonian_bk = of.bravyi_kitaev(hamiltonian_ferm_op)
# Convert to Scipy sparse matrix
hamiltonian_bk_sparse = of.get_sparse_operator(hamiltonian_bk)
# Compute ground state energy
eigs, _ = linalg.eigsh(hamiltonian_bk_sparse, k=1, which="SA")
ground_energy = eigs[0]
print("Ground_energy: {}".format(ground_energy))
print("BKT transformed Hamiltonian:")
print(hamiltonian_bk)
Ground_energy: -1.134147666677097 BKT transformed Hamiltonian: (-0.16733398905695201+0j) [] + (0.04615669588901533+0j) [X0 Z1 X2] + (0.04615669588901533+0j) [X0 Z1 X2 Z3] + (0.04615669588901533+0j) [Y0 Z1 Y2] + (0.04615669588901533+0j) [Y0 Z1 Y2 Z3] + (0.16251648748871642+0j) [Z0] + (0.16251648748871636+0j) [Z0 Z1] + (0.1633603430909739+0j) [Z0 Z1 Z2] + (0.1633603430909739+0j) [Z0 Z1 Z2 Z3] + (0.11720364720195856+0j) [Z0 Z2] + (0.11720364720195856+0j) [Z0 Z2 Z3] + (0.1658325372159041+0j) [Z1] + (-0.19744293699755838+0j) [Z1 Z2 Z3] + (0.17169788392286722+0j) [Z1 Z3] + (-0.1974429369975584+0j) [Z2]
Exercise
- The BCS mean-field d-wave model of superconductivity has the Hamiltonian
\[ H = - t \sum_{\langle i,j \rangle} \sum_\sigma (a^\dagger_{i, \sigma} a_{j, \sigma} + a^\dagger_{j, \sigma} a_{i, \sigma}) - \sum_{\langle i,j \rangle} \Delta_{ij} (a^\dagger_{i, \uparrow} a^\dagger_{j, \downarrow} - a^\dagger_{i, \downarrow} a^\dagger_{j, \uparrow} + a_{j, \downarrow} a_{i, \uparrow} - a_{j, \uparrow} a_{i, \downarrow}) \]
Use the mean_field_dwave function to generate an instance of this model with dimensions 10x10.
- Convert the Hamiltonian to a
QubitOperatorwith the JWT. What is the length of the longest Pauli string that appears? - Convert the Hamiltonian to a
QubitOperatorwith the BKT. What is the length of the longest Pauli string that appears? - Convert the Hamiltonian to a
QuadraticHamiltonian. Get its ground energy using theground_energymethod ofQuadraticHamiltonian. What would happen if you tried to compute the ground energy by converting to a sparse matrix?
Hamiltonian simulation with Trotter formulas
- Goal: apply \(\exp(-i H t)\) where \(H = \sum_j H_j\)
- Use an approximation such as \(\exp(-i H t) \approx (\prod_{j=1} \exp(-i H_j t/r))^r\)
- Exposed via the
openfermion.simulate_trotterfunction - Currently implemented algorithms are from arXiv:1706.00023, arXiv:1711.04789, and arXiv:1808.02625, and are based on the JWT
- Currently supported Hamiltonian types:
DiagonalCoulombHamiltonianandInteractionOperator
As a demonstration, we'll simulate time evolution under the hydrogen molecule Hamiltonian we generated earlier.
First, let's create a random initial state and apply the exact time evolution by matrix exponentiation:
\[ \lvert \psi \rangle \mapsto \exp(-i H t) \lvert \psi \rangle \]
# Create a random initial state
n_qubits = of.count_qubits(hamiltonian)
initial_state = of.haar_random_vector(2**n_qubits, seed=7)
# Set evolution time
time = 1.0
# Apply exp(-i H t) to the state
exact_state = linalg.expm_multiply(-1j * hamiltonian_jw_sparse * time, initial_state)
Now, let's create a circuit to perform the evolution and compare the fidelity of the resulting state with the one from exact evolution. The fidelity can be increased by increasing the number of Trotter steps. Note that the Hamiltonian input to openfermion.simulate_trotter should be an InteractionOperator, not a FermionOperator.
# Initialize qubits
qubits = cirq.LineQubit.range(n_qubits)
# Create circuit
circuit = cirq.Circuit(
of.simulate_trotter(
qubits, hamiltonian, time, n_steps=10, order=0, algorithm=of.LOW_RANK
)
)
# Apply the circuit to the initial state
result = circuit.final_state_vector(initial_state)
# Compute the fidelity with the final state from exact evolution
fidelity = abs(np.dot(exact_state, result.conj())) ** 2
print(fidelity)
0.9999820449924582
print(circuit.to_text_diagram(transpose=True))
0 1 2 3 │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.052π) Rz(0.052π) Rz(0.03π) Rz(0.03π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.022 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.021 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.01π) @──────────────────@^-0.021 Rz(-0.011π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.01π) Rz(-0.011π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.006 │ │ │ │ @─────────────@^-0.006 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(π) ×──────────────────× Rz(0) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(0) Rz(π) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^0 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^0 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.03π) Rz(0) Rz(0) Rz(0.052π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.03π) Rz(0.052π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.022 │ │ │ │ @─────────────@^-0.021 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.011π) @──────────────────@^-0.021 Rz(-0.01π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.011π) Rz(-0.01π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.006 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.006 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(0) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(π) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^0 │ │ │ │ @─────────────@^0 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.052π) Rz(0) Rz(0) Rz(0.03π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.052π) Rz(0.03π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.022 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.021 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.01π) @──────────────────@^-0.021 Rz(-0.011π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.01π) Rz(-0.011π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.006 │ │ │ │ @─────────────@^-0.006 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(π) ×──────────────────× Rz(0) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(0) Rz(π) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^0 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^0 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.03π) Rz(0) Rz(0) Rz(0.052π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.03π) Rz(0.052π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.022 │ │ │ │ @─────────────@^-0.021 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.011π) @──────────────────@^-0.021 Rz(-0.01π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.011π) Rz(-0.01π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.006 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.006 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(0) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(π) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^0 │ │ │ │ @─────────────@^0 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.052π) Rz(0) Rz(0) Rz(0.03π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.052π) Rz(0.03π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.022 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.021 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.01π) @──────────────────@^-0.021 Rz(-0.011π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.01π) Rz(-0.011π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.006 │ │ │ │ @─────────────@^-0.006 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(π) ×──────────────────× Rz(0) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(0) Rz(π) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^0 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^0 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.03π) Rz(0) Rz(0) Rz(0.052π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.03π) Rz(0.052π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.022 │ │ │ │ @─────────────@^-0.021 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.011π) @──────────────────@^-0.021 Rz(-0.01π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.011π) Rz(-0.01π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.006 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.006 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(0) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(π) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^0 │ │ │ │ @─────────────@^0 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.052π) Rz(0) Rz(0) Rz(0.03π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.052π) Rz(0.03π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.022 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.021 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.01π) @──────────────────@^-0.021 Rz(-0.011π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.01π) Rz(-0.011π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.006 │ │ │ │ @─────────────@^-0.006 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(π) ×──────────────────× Rz(0) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(0) Rz(π) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^0 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^0 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.03π) Rz(0) Rz(0) Rz(0.052π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.03π) Rz(0.052π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.022 │ │ │ │ @─────────────@^-0.021 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.011π) @──────────────────@^-0.021 Rz(-0.01π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.011π) Rz(-0.01π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.006 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.006 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(0) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(π) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^0 │ │ │ │ @─────────────@^0 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.052π) Rz(0) Rz(0) Rz(0.03π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.052π) Rz(0.03π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.022 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.021 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.01π) @──────────────────@^-0.021 Rz(-0.011π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.01π) Rz(-0.011π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.006 │ │ │ │ @─────────────@^-0.006 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(π) ×──────────────────× Rz(0) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(0) Rz(π) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^0 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^0 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ Rz(0.03π) Rz(0) Rz(0) Rz(0.052π) │ │ │ │ Rz(0) Rz(0) Rz(0) Rz(0) │ │ │ │ │ Rz(0.03π) Rz(0.052π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-1 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^-1 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^-0.022 │ │ │ │ @─────────────@^-0.021 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^-0.021 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^-0.021 @────────────────@^-0.021 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.011π) @──────────────────@^-0.021 Rz(-0.01π) │ │ │ │ Rz(π) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.011π) Rz(-0.01π) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 Z^0 │ │ │ │ @─────────────@^-0.006 Z^0 │ │ │ │ │ ×─────────────× @────────────────@^-0.006 │ │ │ │ │ │ ×────────────────× │ │ │ │ │ @──────────────────@^0.006 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0.006 @────────────────@^0.006 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(-0.003π) @──────────────────@^0.006 Rz(-0.003π) │ │ │ │ Rz(0) ×──────────────────× Rz(π) │ │ │ │ │ Rz(-0.003π) Rz(-0.003π) │ │ │ │ │ │ Rz(π) Rz(0) │ │ │ │ │ │ PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 PhISwap(0.25)────PhISwap(0.25)^-0.5 │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.5 Z^0 │ │ │ │ │ Z^0 PhISwap(0.25)──────PhISwap(0.25)^-1 │ │ │ │ │ │ Z^0 @────────────────@^0 │ │ │ │ @─────────────@^0 ×────────────────× │ │ │ │ ×─────────────× │ │ │ │ │ │ │ @──────────────────@^0 │ │ │ │ │ │ ×──────────────────× │ │ │ │ │ @─────────────@^0 @────────────────@^0 │ │ │ │ ×─────────────× ×────────────────× │ │ │ │ Rz(0) @──────────────────@^0 Rz(0) │ │ │ │ Rz(0) ×──────────────────× Rz(0) │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │ │ Rz(0) Rz(0) │ │ │ │ │
Bogoliubov transformation
- Single-particle orbital basis change
- In the particle-conserving case, takes the form
\[ U a_p^\dagger U^\dagger = b_p^\dagger, \quad b_p^\dagger = \sum_{q} u_{pq} a_q^\dagger \]
and \(u\) is unitary.
- Can be used to diagonalize any quadratic Hamiltonian:
\[ \sum_{p, q} T_{pq} a_p^\dagger a_q \mapsto \sum_{j} \varepsilon_j b_j^\dagger b_j + \text{constant} \]
- Implementation from arXiv:1711.05395; uses linear depth and linear connectivity
As an example, we'll prepare the ground state of a random particle-conserving quadratic Hamiltonian.
n_qubits = 5
quad_ham = of.random_quadratic_hamiltonian(
n_qubits, conserves_particle_number=True, seed=7
)
print(of.get_fermion_operator(quad_ham))
1.690525703800356 [] + (0.5315776978980016+0j) [0^ 0] + (-1.347208023348913+2.7004721387490935j) [0^ 1] + (-0.28362365442898696-1.8784499457335426j) [0^ 2] + (0.12594647819298657-1.3106154125325498j) [0^ 3] + (-0.3880303291443195-1.1751249212322041j) [0^ 4] + (-1.347208023348913-2.7004721387490935j) [1^ 0] + (2.5012533818678193+0j) [1^ 1] + (0.3391421007279024-3.8305756810505094j) [1^ 2] + (-0.3509690502067961+0.090677856754656j) [1^ 3] + (1.8575239595653907-1.4736314761076197j) [1^ 4] + (-0.28362365442898696+1.8784499457335426j) [2^ 0] + (0.3391421007279024+3.8305756810505094j) [2^ 1] + (-0.019560786804260433+0j) [2^ 2] + (-2.979765944360631+2.5490724453105917j) [2^ 3] + (0.5091942820312417+0.344618218148502j) [2^ 4] + (0.12594647819298657+1.3106154125325498j) [3^ 0] + (-0.3509690502067961-0.090677856754656j) [3^ 1] + (-2.979765944360631-2.5490724453105917j) [3^ 2] + (3.767336752913784+0j) [3^ 3] + (-1.2963431636902167-1.5288970105744286j) [3^ 4] + (-0.3880303291443195+1.1751249212322041j) [4^ 0] + (1.8575239595653907+1.4736314761076197j) [4^ 1] + (0.5091942820312417-0.344618218148502j) [4^ 2] + (-1.2963431636902167+1.5288970105744286j) [4^ 3] + (-0.3445183403145406+0j) [4^ 4]
Now we construct a circuit which maps computational basis states to eigenstates of the Hamiltonian.
_, basis_change_matrix, _ = quad_ham.diagonalizing_bogoliubov_transform()
qubits = cirq.LineQubit.range(n_qubits)
circuit = cirq.Circuit(of.bogoliubov_transform(qubits, basis_change_matrix))
print(circuit.to_text_diagram(transpose=True))
0 1 2 3 4 │ │ │ │ │ Rz(0) Rz(0) Rz(π) Rz(-π) Rz(0) │ │ │ │ │ │ │ │ PhISwap(0.25)─────────PhISwap(0.25)^0.073 │ │ │ │ │ │ │ PhISwap(0.25)────────PhISwap(0.25)^(5/11) Z^(-7/9) │ │ │ │ │ │ PhISwap(0.25)───────PhISwap(0.25)^0.814 Z^-0.508 │ │ │ │ │ │ PhISwap(0.25)─PhISwap(0.25)^0.703 Z^-0.304 PhISwap(0.25)─────────PhISwap(0.25)^0.381 │ │ │ │ │ │ Z^-0.767 PhISwap(0.25)────────PhISwap(0.25)^(13/16) Z^0.401 │ │ │ │ │ │ PhISwap(0.25)───────PhISwap(0.25)^(7/11) Z^(-13/14) │ │ │ │ │ │ │ │ Z^0.393 PhISwap(0.25)─────────PhISwap(0.25)^0.481 │ │ │ │ │ │ │ PhISwap(0.25)────────PhISwap(0.25)^0.422 Z^-0.738 │ │ │ │ │ │ │ │ Z^-0.824 │ │ │ │ │ │ │ │ │ PhISwap(0.25)─────────PhISwap(0.25)^0.506 │ │ │ │ │ │ │ │ │ Z^(5/7) │ │ │ │ │
In the rotated basis, the quadratic Hamiltonian takes the form
\[ H = \sum_j \varepsilon_j b_j^\dagger b_j + \text{constant} \]
We can get the \(\varepsilon_j\) and the constant using the orbital_energies method of QuadraticHamiltonian.
orbital_energies, constant = quad_ham.orbital_energies()
print(orbital_energies)
print(constant)
[-6.25377614 -1.2291963 0.71202361 5.0062515 8.20078604] 1.690525703800356
The ground state of the Hamiltonian is prepared by filling in the orbitals with negative energy.
# Apply the circuit with initial state having the first two modes occupied.
result = circuit.final_state_vector(initial_state=0b11000)
# Compute the expectation value of the final state with the Hamiltonian
quad_ham_sparse = of.get_sparse_operator(quad_ham)
print(of.expectation(quad_ham_sparse, result))
# Print out the ground state energy; it should match
print(quad_ham.ground_energy())
(-5.792446738060052+1.1102230246251565e-16j) -5.792446738060049
Recall that the Jordan-Wigner transform of \(b_j^\dagger b_j\) is \(\frac12(I-Z)\). Therefore, \(\exp(-i \varepsilon_j b_j^\dagger b_j)\) is equivalent to a single-qubit Z rotation under the JWT. Since the operators \(b_j^\dagger b_j\) commute, we have
\[ \exp(-i H t) = \exp(-i \sum_j \varepsilon_j b_j^\dagger b_j t) = \prod_j \exp(-i \varepsilon_j b_j^\dagger b_j t) \]
This gives a method for simulating time evolution under a quadratic Hamiltonian:
- Use a Bogoliubov transformation to change to the basis in which the Hamiltonian is diagonal (Note: this transformation might be the inverse of what you expect. In that case, use
cirq.inverse) - Apply single-qubit Z-rotations with angles proportional to the orbital energies
- Undo the basis change
The code cell below creates a random initial state and applies time evolution by direct matrix exponentiation.
# Create a random initial state
initial_state = of.haar_random_vector(2**n_qubits)
# Set evolution time
time = 1.0
# Apply exp(-i H t) to the state
final_state = linalg.expm_multiply(-1j * quad_ham_sparse * time, initial_state)
Exercise
Fill in the code cell below to construct a circuit which applies \(\exp(-i H t)\) using the method described above
# Initialize qubits
qubits = cirq.LineQubit.range(n_qubits)
# Write code below to create the circuit
# You should define the `circuit` variable here
# ---------------------------------------------
# ---------------------------------------------
# Apply the circuit to the initial state
result = circuit.final_state_vector(initial_state)
# Compute the fidelity with the correct final state
fidelity = abs(np.dot(final_state, result.conj())) ** 2
# Print fidelity; it should be 1
print(fidelity)
0.08926042490120051
Solution
# Initialize qubits
qubits = cirq.LineQubit.range(n_qubits)
# Write code below to create the circuit
# You should define the `circuit` variable here
# ---------------------------------------------
def exponentiate_quad_ham(qubits, quad_ham):
_, basis_change_matrix, _ = quad_ham.diagonalizing_bogoliubov_transform()
orbital_energies, _ = quad_ham.orbital_energies()
yield cirq.inverse(of.bogoliubov_transform(qubits, basis_change_matrix))
for i in range(len(qubits)):
yield cirq.rz(rads=-orbital_energies[i]).on(qubits[i])
yield of.bogoliubov_transform(qubits, basis_change_matrix)
circuit = cirq.Circuit(exponentiate_quad_ham(qubits, quad_ham))
# ---------------------------------------------
# Apply the circuit to the initial state
result = circuit.final_state_vector(initial_state)
# Compute the fidelity with the correct final state
fidelity = abs(np.dot(final_state, result.conj())) ** 2
# Print fidelity; it should be 1
print(fidelity)
0.999999999999994