Distributions of Noisy Expectation Values over Sets of Measurement Operators

TL;DR

This paper analyzes the distribution of expectation values of measurement operators in quantum systems, deriving moments, and comparing empirical data from noisy quantum circuits to effective models.

Contribution

It generalizes previous work to sets of measurement operators and introduces an effective distribution model for noisy quantum circuit expectation values.

Findings

Effective distribution models reproduce peak behavior across various parameters.

Deviations in tails are due to local noise effects.

Measurement operator symmetry affects the distribution modality.

Abstract

Expectation values of measurement operators, interpreted as measurement probabilities, arise frequently throughout quantum algorithms. When quantum states are randomly distributed, their expectation values are also randomly distributed. In this work, with the goal of understanding non-unitary dynamics, we generalize previous derivations for distributions of expectation values (Campos Venuti and Zanardi, Physics Letters A (377), 2013) to the case of sets of measurement operators and random mixed quantum states within variable sized environments. Using combinatorics approaches, we derive expressions for their moments. We proceed to construct empirical distributions of simulated Haar random brickwork quantum circuits with local depolarizing noise, and compare their form to a proposed effective global-depolarizing-like model with variable effective noise scales and environment dimensions. The fitted effective distributions reproduce peak behaviour across circuit depths, noise scales, and system sizes, while deviations in the distribution tails arise from local noise effects. The fit effective model parameters are also shown to vary smoothly and consistently with circuit depth and noise scale. Finally, sets of non-symmetric measurement operators are shown to exhibit distinct multi-modal distributions relative to uni-modal distributions for symmetric measurement operators, opening up questions about their simulability.