Matchgate circuit representation of fermionic Gaussian states: optimal preparation, approximation, and classical simulation

TL;DR

This paper investigates optimal methods for preparing fermionic Gaussian states using matchgate circuits, establishes bounds on gate requirements, and introduces a classical simulation algorithm for these quantum states.

Contribution

It provides the first explicit optimal algorithms for FGS preparation with matchgate circuits and a new classical simulation method based on generating circuits.

Findings

Derived lower bounds on gate counts for FGS preparation.

Presented algorithms that saturate these bounds, proving optimality.

Developed a classical simulation algorithm for matchgate circuits.

Abstract

Fermionic Gaussian states (FGSs) and the associated matchgate circuits play a central role in quantum information theory and condensed matter physics. Despite being possibly highly entangled, they can still be efficiently simulated on classical computers. We address the question of how to optimally create such states when using matchgate circuits acting on product states. To this end, we derive lower bounds on the number of gates required to prepare an arbitrary pure FGS: We establish both an asymptotic bound on the minimal gate count over general nearest-neighbor gate sets and an exact bound for circuits composed solely of matchgates. We present explicit algorithms whose constructions saturate these bounds, thereby proving their optimality. We furthermore determine when an FGS can be prepared with a circuit of any given depth, and derive an algorithm that constructs such a circuit whenever this condition is satisfied, either exactly or approximately. Our results have direct applications to (approximate) state preparation and to disentangling procedures. Moreover, we introduce a new classical simulation algorithm for matchgate circuits, based entirely on manipulating the generating circuits of the FGSs. Finally, we briefly study an extension of our framework for $t$-doped Gaussian states and circuits.