Classical simulation of non-Gaussian fermionic circuits

TL;DR

This paper introduces efficient classical algorithms for simulating non-Gaussian fermionic circuits by extending covariance matrix formalism, enabling polynomial-time simulation based on the initial state's non-Gaussianity.

Contribution

It develops a novel extension of the covariance matrix formalism to simulate non-Gaussian fermionic states efficiently, connecting to Clifford circuit simulation techniques.

Findings

Simulation algorithms have polynomial complexity in fermion number and accuracy.

The fermionic Gaussian extent is multiplicative under tensor products for certain states.

The approach generalizes classical simulation methods for non-Gaussian fermionic systems.

Abstract

We propose efficient algorithms for classically simulating fermionic linear optics operations applied to non-Gaussian initial states. By gadget constructions, this provides algorithms for fermionic linear optics with non-Gaussian operations. We argue that this problem is analogous to that of simulating Clifford circuits with non-stabilizer initial states: Algorithms for the latter problem immediately translate to the fermionic setting. Our construction is based on an extension of the covariance matrix formalism which permits to efficiently track relative phases in superpositions of Gaussian states. It yields simulation algorithms with polynomial complexity in the number of fermions, the desired accuracy, and certain quantities capturing the degree of non-Gaussianity of the initial state. We study one such quantity, the fermionic Gaussian extent, and show that it is multiplicative on tensor products when the so-called fermionic Gaussian fidelity is. We establish this property for the tensor product of two arbitrary pure states of four fermions with positive parity.