Classical Shadows With Noise

TL;DR

This paper analyzes how noise affects the classical shadows protocol for quantum state estimation, providing bounds on sample complexity and proposing a noise-resilient estimator for near-term quantum hardware.

Contribution

It derives analytical bounds on sample complexity under noise and introduces an unbiased estimator that mitigates noise effects in the classical shadows protocol.

Findings

Sample complexity bounds derived for local and global noise.

A new unbiased estimator for noisy quantum measurements.

Application of results to depolarizing and amplitude damping noise.

Abstract

The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.