The computational power of matchgates and the XY interaction on arbitrary graphs

TL;DR

This paper characterizes the computational capabilities of matchgates and XY interactions on various graphs, showing universality on most graphs except paths and cycles where they are classically simulable.

Contribution

It provides a complete classification of the computational power of matchgates and XY interactions on arbitrary graphs, extending previous results.

Findings

Matchgates are universal on all connected graphs except paths and cycles.

XY interaction shares the same computational dichotomy as matchgates.

Classical simulability holds specifically for cycles.

Abstract

Matchgates are a restricted set of two-qubit gates known to be classically simulable when acting on nearest-neighbor qubits on a path, but universal for quantum computation when the qubits are arranged on certain other graphs. Here we characterize the power of matchgates acting on arbitrary graphs. Specifically, we show that they are universal on any connected graph other than a path or a cycle, and that they are classically simulable on a cycle. We also prove the same dichotomy for the XY interaction, a proper subset of matchgates related to some implementations of quantum computing.