A Quantum Algorithm for Functions of Multiple Commuting Hermitian Matrices

TL;DR

This paper introduces a quantum algorithm for transforming functions of multiple commuting Hermitian matrices, enabling efficient computation of polynomial matrix functions and functions of normal matrices on quantum computers.

Contribution

It develops a multivariate quantum eigenvalue transformation framework for functions of commuting Hermitian matrices, extending quantum signal processing capabilities.

Findings

Framework for polynomial matrix functions

Application to functions of normal matrices

Enables quantum computation of matrix functions

Abstract

Quantum signal processing allows for quantum eigenvalue transformation with Hermitian matrices, in which each eigenspace component of an input vector gets transformed according to its eigenvalue. In this work, we introduce the multivariate quantum eigenvalue transformation for functions of commuting Hermitian matrices. We then present a framework for working with polynomial matrix functions in which we may solve MQET, and give the application of computing functions of normal matrices using a quantum computer.