Performance analysis of multi-shot shadow estimation

TL;DR

This paper analyzes the performance of multi-shot shadow estimation in quantum state measurement, introducing a new cross-shadow-norm to better understand variance and providing bounds and formulas for different measurement types.

Contribution

It introduces the cross-shadow-norm to analyze variance in multi-shot shadow estimation and derives bounds and exact formulas for specific measurement scenarios.

Findings

Variance is related to the shadow-norm and the new cross-shadow-norm.

Upper bounds for the cross-shadow-norm are established for Pauli and Clifford measurements.

Exact variance formula derived for Pauli observables under random Pauli measurements.

Abstract

Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for $K$ times after the same unitary evolution, and repeats this procedure for $M$ rounds of random sampled unitary. As a result, there are $MK$ times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable $O$. We find that in addition to the shadow-norm $\|O \|_{\mathrm{shadow}}$ introduced in [Huang et.al.~Nat.~Phys.~2020\cite{huang2020predicting}], the variance is also related to another norm, and we denote it as the cross-shadow-norm $\|O \|_{\mathrm{Xshadow}}$. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of $\|O \|_{\mathrm{Xshadow}}$. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.