Displaced Fermionic Gaussian States and their Classical Simulation

TL;DR

This paper introduces a comprehensive framework for displaced fermionic Gaussian states, providing efficient classical simulation methods, equivalence characterizations, and a novel embedding to extend Gaussian testing protocols, advancing fermionic quantum computation analysis.

Contribution

It presents a unified characterization of displaced fermionic Gaussian states, an efficient simulation protocol, and a new embedding to generalize Gaussian testing to displaced states.

Findings

Efficient classical simulation protocol for displaced Gaussian circuits

Equivalence between different characterizations of displaced Gaussian states

A Gaussianity-preserving embedding for state analysis

Abstract

This work explores displaced fermionic Gaussian operators with nonzero linear terms. We first demonstrate equivalence between several characterizations of displaced Gaussian states. We also provide an efficient classical simulation protocol for displaced Gaussian circuits and demonstrate their computational equivalence to circuits composed of nearest-neighbor matchgates augmented by single-qubit gates on the initial line. Finally, we construct a novel Gaussianity-preserving unitary embedding that maps $n$-qubit displaced Gaussian states to $(n+1)$-qubit even Gaussian states. This embedding facilitates the generalization of existing Gaussian testing protocols to displaced Gaussian states and unitaries. Our results provide new tools to analyze fermionic systems beyond the constraints of parity super-selection, extending the theoretical understanding and practical simulation of fermionic quantum computation.