Implementing smooth functions of a hermitian matrix on a quantum computer

Abstract

We consider 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 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 (Formula Presented) where ∈ 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).