Quantum matchgate computations and linear threshold gates

TL;DR

This paper characterizes the class of boolean functions computable by matchgate circuits, showing they are exactly the linear threshold gates, and reveals their limited power in high-success probability scenarios.

Contribution

It provides a complete characterization of matchgate-computable functions as linear threshold gates and analyzes their computational power in high-success regimes.

Findings

Matchgate circuits compute exactly linear threshold functions.

High success probability restricts matchgate functions to trivial cases.

Matchgate power is limited when success probability exceeds 3/4.

Abstract

The theory of matchgates is of interest in various areas in physics and computer science. Matchgates occur in e.g. the study of fermions and spin chains, in the theory of holographic algorithms and in several recent works in quantum computation. In this paper we completely characterize the class of boolean functions computable by unitary two-qubit matchgate circuits with some probability of success. We show that this class precisely coincides with that of the linear threshold gates. The latter is a fundamental family which appears in several fields, such as the study of neural networks. Using the above characterization, we further show that the power of matchgate circuits is surprisingly trivial in those cases where the computation is to succeed with high probability. In particular, the only functions that are matchgate-computable with success probability greater than 3/4 are functions depending on only a single bit of the input.