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

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