Fermionic Linear Optics and Matchgates

TL;DR

This paper analyzes fermionic linear optics, showing its limitations and relation to matchgates, and explains why it is classically simulatable, contrasting it with bosonic systems.

Contribution

It demonstrates that fermionic linear optics states form the closure of a low-dimensional Lie group and links matchgates to extended fermionic linear optics operators.

Findings

Fermionic linear optics is classically efficiently simulatable.

Matchgates generate a monoid of extended fermionic linear optics operators.

The set of achievable states forms the closure of a low-dimensional Lie group.

Abstract

Fermionic linear optics is efficiently classically simulatable. Here it is shown that the set of states achievable with fermionic linear optics and particle measurements is the closure of a low dimensional Lie group. The weakness of fermionic linear optics and measurements can therefore be explained and contrasted with the strength of bosonic linear optics with particle measurements. An analysis of fermionic linear optics is used to show that the two-qubit matchgates and the simulatable matchcircuits introduced by Valiant generate a monoid of extended fermionic linear optics operators. A useful interpretation of efficient classical simulations such as this one is as a simulation of a model of non-deterministic quantum computation. Problem areas for future investigations are suggested.