Measuring Non-Gaussian Magic in Fermions: Convolution, Entropy, and the Violation of Wick's Theorem and the Matchgate Identity

TL;DR

This paper introduces measures of non-Gaussian magic in fermionic quantum states, demonstrating convergence to Gaussian states via convolution and using violations of Wick's theorem and matchgate identities for quantification.

Contribution

It identifies convolution as a tool for fermions, proves a central limit theorem for fermionic systems, and links non-Gaussian magic to fundamental quantum properties.

Findings

Three notions of Gaussification coincide for fermions.

Efficient measures of non-Gaussian magic are developed.

Violations of Wick's theorem quantify non-Gaussian magic.

Abstract

Classically hard to simulate quantum states, or "magic states", are prerequisites to quantum advantage, highlighting an apparent separation between classically and quantumly tractable problems. Classically simulable states such as Clifford circuits on stabilizer states, free bosonic states, free fermions, and matchgate circuits are all in some sense Gaussian. While free bosons and fermions arise from quadratic Hamiltonians, recent works have demonstrated that bosonic and qudit systems converge to Gaussians and stabilizers under convolution. In this work, we similarly identify convolution for fermions and find efficient measures of non-Gaussian magic in pure fermionic states. We demonstrate that three natural notions for the Gaussification of a state, (1) the Gaussian state with the same covariance matrix, (2) the fixed point of convolution, and (3) the closest Gaussian in relative entropy, coincide by proving a central limit theorem for fermionic systems. We then utilize the violation of Wick's theorem and the matchgate identity to quantify non-Gaussian magic in addition to a SWAP test.