Trimmed Sampling Algorithm for the Noisy Generalized Eigenvalue Problem

TL;DR

The paper introduces a Bayesian-based trimmed sampling algorithm to improve the accuracy and reliability of solving noisy generalized eigenvalue problems in large quantum systems, outperforming standard regularization methods.

Contribution

It presents a novel Bayesian inference framework that effectively handles errors in matrix elements, providing more reliable eigenvector and observable estimates in noisy conditions.

Findings

Outperforms standard regularization methods in noisy scenarios

Provides reliable error estimates for eigenvectors and observables

Applicable to classical and quantum large-scale systems

Abstract

Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. It uses projection onto a set of basis states which are typically not orthogonal. One needs to invert a matrix whose entries are inner products of the basis states, and the process is unfortunately susceptible to even small errors. The problem is especially bad when matrix elements are evaluated using stochastic methods and have significant error bars. In this work, we introduce the trimmed sampling algorithm in order to solve this problem. Using the framework of Bayesian inference, we sample prior probability distributions determined by uncertainty estimates of the various matrix elements and likelihood functions composed of physics-informed constraints. The result is a probability distribution for the eigenvectors and observables which automatically comes with a reliable estimate of the error and performs far better than standard regularization methods. The method should have immediate use for a wide range of applications involving classical and quantum computing calculations of large quantum systems.