Nonstabilizerness Enhances Thrifty Shadow Estimation

TL;DR

This paper demonstrates that thrifty shadow estimation is effective with 2-design unitaries, and reveals how nonstabilizerness reduces variance, providing new insights and practical methods for quantum state fidelity estimation in NISQ devices.

Contribution

It establishes the effectiveness of thrifty shadow estimation with 2-designs and links nonstabilizerness to variance reduction, introducing a simple circuit for NISQ-era applications.

Findings

Thrifty shadow estimation is effective on average with 2-design unitaries.

Variance inversely correlates with nonstabilizerness of states and observables.

Fidelity estimation variance decreases exponentially with stabilizer 2-Rényi entropy.

Abstract

Shadow estimation is a powerful approach for estimating the expectation values of many observables. Thrifty shadow estimation is a simple variant that is proposed to reduce the experimental overhead by reusing random circuits repeatedly. Although this idea is so simple, its performance is quite elusive. In this work we show that thrifty shadow estimation is effective on average whenever the unitary ensemble forms a 2-design, in sharp contrast with the previous expectation. In thrifty shadow estimation based on the Clifford group, the variance is inversely correlated with the degree of nonstabilizerness of the state and observable, which is a key resource in quantum information processing. For fidelity estimation, it decreases exponentially with the stabilizer 2-R\'{e}nyi entropy of the target state, which endows the stabilizer 2-R\'{e}nyi entropy with a clear operational meaning. In addition,we propose a simple circuit to enhance the efficiency, which requires only one layer of $T$ gates and is particularly appealing in the NISQ era.