Quantum algorithm for matrix functions by Cauchy's integral formula

TL;DR

This paper introduces a quantum algorithm for computing functions of matrices applied to vectors, leveraging Cauchy's integral formula to achieve logarithmic dependence on the desired accuracy and handle non-Hermitian matrices.

Contribution

The authors develop a quantum algorithm that avoids eigenvalue estimation, enabling efficient computation of matrix functions for non-Hermitian matrices with improved accuracy dependence.

Findings

Runtime is polynomial in log(1/ε)

Algorithm works for non-Hermitian matrices

Avoids eigenvalue estimation using Cauchy's integral formula

Abstract

For matrix $A$, vector $\boldsymbol{b}$ and function $f$, the computation of vector $f(A)\boldsymbol{b}$ arises in many scientific computing applications. We consider the problem of obtaining quantum state $\lvert f \rangle$ corresponding to vector $f(A)\boldsymbol{b}$. There is a quantum algorithm to compute state $\lvert f \rangle$ using eigenvalue estimation that uses phase estimation and Hamiltonian simulation $\mathrm{e}^{\mathrm{{\bf i}} A t}$. However, the algorithm based on eigenvalue estimation needs $\textrm{poly}(1/\epsilon)$ runtime, where $\epsilon$ is the desired accuracy of the output state. Moreover, if matrix $A$ is not Hermitian, $\mathrm{e}^{\mathrm{{\bf i}} A t}$ is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is $\mathrm{poly}(\log(1/\epsilon))$ and the algorithm outputs state $\lvert f \rangle$ even if $A$ is not Hermitian.