openfermioncirq.trotter.LowRankTrotterAlgorithm

class openfermioncirq.trotter.LowRankTrotterAlgorithm(truncation_threshold: Optional[float] = 1e-08, final_rank: Optional[int] = None, spin_basis=True)

A Trotter algorithm using the low rank decomposition strategy.

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 \(\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s = \sum_{j=0}^{J-1} \lambda_j (\sum_{pq} g_{jpq} a^\dagger_p a_q)^2\) 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 \(\prod_{j=0}^{J-1} R_j e^{-i \lambda_j \sum_{pq} f_{pj} f_{qj} n_p n_q} R_j^\dagger\). The \(R_j\) are Bogoliubov transformations, and one can use a swap network to simulate the diagonal \(n_p n_q\) 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 \(\sum_{l=0}^{L-1} (\sum_{pq} |g_{lpq}|)^2 |\lambda_l| < x\) where x is a truncation threshold specified by user.

__init__(truncation_threshold: Optional[float] = 1e-08, final_rank: Optional[int] = None, spin_basis=True) → None

Parameters:

• truncation_threshold – The value of x from the docstring of this class.
• final_rank – If provided, this specifies the value of J at which to truncate.
• spin_basis – Whether the Hamiltonian is given in the spin orbital (rather than spatial orbital) basis.

Methods

__init__(truncation_threshold, final_rank[, …])	param truncation_threshold:  The value of x from the docstring of
asymmetric(hamiltonian, …)
controlled_asymmetric(hamiltonian, …)
controlled_symmetric(hamiltonian, …)
symmetric(hamiltonian, …)

Attributes

supported_types