openfermion.circuits.LowRankTrotterAlgorithm

A Trotter algorithm using the low rank decomposition strategy.

Inherits From: TrotterAlgorithm

Used in the notebooks

Used in the tutorials

This algorithm simulates an InteractionOperator with real coefficients. The one-body terms are simulated in their diagonal basis; the basis change is accomplished with a Bogoliubov transformation. To simulate the two-body terms, the two-body tensor is decomposed into singular components and possibly truncating. Then, each singular component is simulated in the appropriate basis using a (non-fermionic) swap network. The general idea is based on expressing the two-body operator as :math:\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s = \sum_{j=0}^{J-1} \lambda_j (\sum_{pq} g_{jpq} a^\dagger_p a_q)^2 One can then diagonalize the squared one-body component as math:\sum_{pq} g_{pqj} a^\dagger_p a_q = R_j (\sum_{p} f_{pj} n_p) R_j^\dagger Then, a 'low rank' Trotter step of the two-body tensor can be simulated as :math:\prod_{j=0}^{J-1} R_j e^{-i \lambda_j \sum_{pq} f_{pj} f_{qj} n_p n_q} R_j^\dagger. The :math:R_j are Bogoliubov transformations, and one can use a swap network to simulate the diagonal :math:n_p n_q terms. The value of J is either fully the square of the number of qubits, which would imply no truncation, or it is specified by the user, or it is chosen so that :math:\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x where x is a truncation threshold specified by user.

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