Extending Classically Simulatable Bounds of Clifford Circuits with Nonstabilizer States via Framed Wigner Functions

TL;DR

This paper introduces a new classical simulation method for qubit Clifford circuits using framed Wigner functions, enabling efficient sampling of certain nonstabilizer states and outcomes without negativity issues.

Contribution

It extends the Wigner function formalism with a phase degree of freedom, allowing Clifford gates to be simulated classically with nonstabilizer states.

Findings

Efficient polynomial-time sampling of some nonstabilizer outcomes

Graph-theoretical method to identify simulatable outcomes

Application to log-depth random Clifford circuits

Abstract

The Wigner function formalism has played a pivotal role in examining the non-classical aspects of quantum states and their classical simulatability. Nevertheless, its application in qubit systems faces limitations due to negativity induced by Clifford gates. In this work, we propose a novel classical simulation method for qubit Clifford circuits based on the framed Wigner function, an extended form of the Wigner function with an additional phase degree of freedom. In our framework, Clifford gates do not induce negativity by switching to a suitable frame; thereby, a wide class of nonstabilizer states can be represented positively. By leveraging this technique, we show that some marginal outcomes of Clifford circuits with nonstabilizer state inputs can be efficiently sampled at polynomial time and memory costs. We develop a graph-theoretical approach to identify classically simulatable marginal outcomes and apply it to log-depth random Clifford circuits. We also present the outcome probability estimation scheme using the framed Wigner function and discuss its precision. Our approach opens new avenues for utilizing quasi-probabilities to explore classically simulatable quantum circuits.