Estimating shots and variance on noisy quantum circuits

TL;DR

This paper introduces a method to estimate the number of measurement shots needed in noisy quantum circuits to reach a target variance, aiding efficient quantum experiment design.

Contribution

It provides a novel approach to predict shot requirements by decomposing variance into statistical and bias components, validated on variational quantum algorithms.

Findings

Accurately predicts variance within known error bounds.

Estimates that 7000 shots suffice for desired variance on IBM Pittsburgh hardware.

Applicable to expectation-value circuits in noisy quantum devices.

Abstract

We present a method for estimating the number of shots required to achieve a desired variance in the results of a quantum circuit. First, we establish a baseline for single-qubit characterisation of individual noise sources. We then move on to multi-qubit circuits, focusing on expectation-value circuits. We decompose the variance of the estimator into a sum of a statistical term and a bias floor. These are independently estimated with one additional run of the circuit. We test our method on a Variational Quantum Eigensolver for $H_2$ and show that we can predict the variance to within known error bounds. We go on to show that for IBM Pittsburgh's noise characteristics, at that instant, 7000 shots for the given circuit would have achieved a $\sigma^2 \approx 0.01$