Foundations for Bayesian inference with engineered likelihood functions for robust amplitude estimation

TL;DR

This paper develops a mathematical framework for robust quantum amplitude estimation using engineered likelihood functions, optimizing Bayesian inference to improve information gain in quantum sampling.

Contribution

It introduces a systematic method for designing and selecting optimal likelihood functions in quantum circuits for amplitude estimation, enhancing Bayesian inference techniques.

Findings

Likelihood functions characterized for specific quantum circuits

Optimal ELF parameters can be systematically chosen

Numerical results show improved estimation performance

Abstract

We present mathematical and conceptual foundations for the task of robust amplitude estimation using engineered likelihood functions (ELFs), a framework introduced in Wang et al. [PRX Quantum 2, 010346 (2021)] that uses Bayesian inference to enhance the rate of information gain in quantum sampling. These ELFs, which are obtained by choosing tunable parameters in a parametrized quantum circuit to minimize the expected posterior variance of an estimated parameter, play an important role in estimating the expectation values of quantum observables. We give a thorough characterization and analysis of likelihood functions arising from certain classes of quantum circuits and combine this with the tools of Bayesian inference to give a procedure for picking optimal ELF tunable parameters. Finally, we present numerical results to demonstrate the performance of ELFs.