Positive Wigner functions render classical simulation of quantum computation efficient

TL;DR

This paper demonstrates that quantum circuits with positive Wigner functions for initial states and operations can be efficiently simulated classically, extending the Gottesman-Knill theorem to continuous and discrete systems.

Contribution

It generalizes the Gottesman-Knill theorem by showing positivity of Wigner functions enables efficient classical simulation of certain quantum circuits.

Findings

Classical simulation is efficient for circuits with positive Wigner functions.

The algorithm can sample from the output distribution, including approximate distributions.

Negativity of the Wigner function is identified as a resource for quantum computational advantage.

Abstract

We show that quantum circuits where the initial state and all the following quantum operations can be represented by positive Wigner functions can be classically efficiently simulated. This is true both for continuous-variable as well as discrete variable systems in odd prime dimensions, two cases which will be treated on entirely the same footing. Noting the fact that Clifford and Gaussian operations preserve the positivity of the Wigner function, our result generalizes the Gottesman-Knill theorem. Our algorithm provides a way of sampling from the output distribution of a computation or a simulation, including the efficient sampling from an approximate output distribution in case of sampling imperfections for initial states, gates, or measurements. In this sense, this work highlights the role of the positive Wigner function as separating classically efficiently simulatable systems from those that are potentially universal for quantum computing and simulation, and it emphasizes the role of negativity of the Wigner function as a computational resource.