Efficient Bayesian phase estimation using mixed priors

TL;DR

This paper presents an efficient Bayesian quantum phase estimation method that dynamically switches between Fourier and normal distribution representations to handle noise and multiple eigenstates, reducing computational complexity.

Contribution

It introduces a novel dynamic switching approach between distribution representations and an efficient weight estimation method for superpositions, improving Bayesian phase estimation performance.

Findings

Reduced update time complexity through analytic expressions

Effective handling of multiple eigenstates with convex optimization

Stable phase distribution representation with bounded errors

Abstract

We describe an efficient implementation of Bayesian quantum phase estimation in the presence of noise and multiple eigenstates. The main contribution of this work is the dynamic switching between different representations of the phase distributions, namely truncated Fourier series and normal distributions. The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. This necessitates truncation of the series, which eventually causes the distribution to become unstable. We derive bounds on the error in representing normal distributions with a truncated Fourier series, and use these to decide when to switch to the normal-distribution representation. This representation is much simpler, and was proposed in conjunction with rejection filtering for approximate Bayesian updates. We show that, in many cases, the update can be done exactly using analytic expressions, thereby greatly reducing the time complexity of the updates. Finally, when dealing with a superposition of several eigenstates, we need to estimate the relative weights. This can be formulated as a convex optimization problem, which we solve using a gradient-projection algorithm. By updating the weights at exponentially scaled iterations we greatly reduce the computational complexity without affecting the overall accuracy.