Implementing smooth functions of a Hermitian matrix on a quantum computer

TL;DR

This paper reviews and analyzes methods for implementing smooth functions of sparse Hermitian matrices on quantum computers, focusing on a combination of techniques that optimize resource use and simplicity.

Contribution

It introduces a new approach combining linear combination of unitaries with Chebyshev polynomial approximations for efficient quantum implementation.

Findings

Query complexity is O(log C/eps) for approximation precision eps.

Success probability depends on the 1-norm of Taylor series coefficients and matrix properties.

The method offers advantages in simplicity and resource consumption in certain cases.

Abstract

We review existing methods for implementing smooth functions f(A) of a sparse Hermitian matrix A on a quantum computer, and analyse a further combination of these techniques which has some advantages of simplicity and resource consumption in some cases. Our construction uses the linear combination of unitaries method with Chebyshev polynomial approximations. The query complexity we obtain is O(log C/eps) where eps is the approximation precision, and C>0 is an upper bound on the magnitudes of the derivatives of the function f over the domain of interest. The success probability depends on the 1-norm of the Taylor series coefficients of f, the sparsity d of the matrix, and inversely on the smallest singular value of the target matrix f(A).