Hermitian Matrix Definiteness from Quantum Phase Estimation

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.

19 pages, 8 figures, 2 algorithms

References in corpus (4)

• Simulating Hamiltonian dynamics with a truncated Taylor series
• A Grand Unification of Quantum Algorithms
• Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits
• Bayesian Quantum Multiphase Estimation Algorithm