openfermion.circuits.low_rank_two_body_decomposition

Convert two-body operator into sum of squared one-body operators.

openfermion.circuits.low_rank_two_body_decomposition(
    two_body_coefficients,
    truncation_threshold=1e-08,
    final_rank=None,
    spin_basis=True
)

As in arXiv:1808.02625, this function decomposes \(\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s\) as \(\sum_{l} \lambda_l (\sum_{pq} g_{lpq} a^\dagger_p a_q)^2\) l is truncated to take max value L so that \(\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x\)

two_body_coefficients ndarray

an N x N x N x N numpy array giving the \(h_{pqrs}\) tensor. This tensor must be 8-fold symmetric (real integrals).

truncation_threshold optional Float

the value of x, above.

final_rank optional int

if provided, this specifies the value of L at which to truncate. This overrides truncation_threshold.

spin_basis bool

True if the two-body terms are passed in spin orbital basis. False if already in spatial orbital basis.

eigenvalues ndarray of floats

length L array giving the \(\lambda_l\).

one_body_squares ndarray of floats

L x N x N array of floats corresponding to the value of \(g_{pql}\).

one_body_correction ndarray

One-body correction terms that result from reordering to chemist ordering, in spin-orbital basis.

truncation_value float

after truncation, this is the value \(\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x\)

TypeError Invalid two-body coefficient tensor specification.