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

Understand

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.