Matchgates and classical simulation of quantum circuits

TL;DR

This paper demonstrates that certain quantum circuits composed of matchgates can be efficiently simulated classically, but slight modifications enable universal quantum computation, highlighting the delicate boundary between classical and quantum computational power.

Contribution

The paper extends the classical simulation results of matchgate circuits and shows how minimal resource additions enable universal quantum computation.

Findings

Nearest neighbor matchgate circuits are classically simulatable.

Allowing next-nearest neighbor interactions enables universal quantum computation.

Generalization to Gaussian quantum circuits expands the class of efficiently simulatable quantum circuits.

Abstract

Let G(A,B) denote the 2-qubit gate which acts as the 1-qubit SU(2) gates A and B in the even and odd parity subspaces respectively, of two qubits. Using a Clifford algebra formalism we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allowing the same gates to act only on n.n. and next-n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan-Wigner representation of a Clifford algebra, we show how one may generalise the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.