Hermitian Matrix Definiteness from Quantum Phase Estimation

TL;DR

This paper introduces a quantum algorithm leveraging phase estimation to classify Hermitian matrices by their definiteness, achieving high accuracy with fewer qubits and improved efficiency for specific matrix classes.

Contribution

It presents a novel quantum algorithm that classifies Hermitian matrices based on their signature using quantum phase estimation, with enhanced performance for certain matrix types.

Findings

Achieves 97% correct classification accuracy.

Computational cost comparable to classical methods for general matrices.

Significant efficiency gains for k-local or sparse Hamiltonians.

Abstract

An algorithm to classify a general Hermitian matrix according to its signature (positive semi-definite, negative or indefinite) is presented. It builds on the Quantum Phase Estimation algorithm, which stores the sign of the eigenvalues of a Hermitian matrix in one ancillary qubit. The signature of the matrix is extracted from the mean value of a spin operator in this single ancillary qubit. The algorithm is probabilistic, but it shows good performance, achieving 97% of correct classifications with few qubits. The computational cost scales comparably to the classical one in the case of a generic matrix, but improves significantly for restricted classes of matrices like $k$-local or sparse hamiltonians.