Quantum Eigenvalue Estimation for Irreducible Non-negative Matrices

TL;DR

This paper presents a quantum eigenvalue estimation method for irreducible non-negative matrices that does not require prior eigenvector knowledge, with success probability estimates based on matrix properties, applicable to various quantum problems.

Contribution

It introduces a quantum phase estimation approach that determines the principal eigenvalue using an equal superposition state, removing the need for an initial eigenvector guess.

Findings

Success probability relates to the operator's closeness to a stochastic matrix

Provides a priori success probability estimates based on column sum variance

Demonstrates applicability to random symmetric matrices and non-negative Hamiltonians

Abstract

Quantum phase estimation algorithm has been successfully adapted as a sub frame of many other algorithms applied to a wide variety of applications in different fields. However, the requirement of a good approximate eigenvector given as an input to the algorithm hinders the application of the algorithm to the problems where we do not have any prior knowledge about the eigenvector. In this paper, we show that the principal eigenvalue of an irreducible non-negative operator can be determined by using an equal superposition initial state in the phase estimation algorithm. This removes the necessity of the existence of an initial good approximate eigenvector. Moreover, we show that the success probability of the algorithm is related to the closeness of the operator to a stochastic matrix. Therefore, we draw an estimate for the success probability by using the variance of the column sums of the operator. This provides a priori information which can be used to know the success probability of the algorithm beforehand for the non-negative matrices and apply the algorithm only in cases when the estimated probability reasonably high. Finally, we discuss the possible applications and show the results for random symmetric matrices and 3-local Hamiltonians with non-negative off-diagonal elements.