Fourier-based quantum signal processing

TL;DR

This paper introduces a Fourier-based quantum signal processing algorithm for Hermitian operators that efficiently implements operator functions using only a single ancilla qubit, improving over previous methods.

Contribution

It presents a novel Fourier approximation algorithm for Hermitian operator functions that requires only one ancilla qubit regardless of the approximation degree.

Findings

Efficient quantum algorithm for Hermitian operator functions.

Classical method for computing Fourier series parameters.

Compatibility with Trotterised and hybrid quantum simulation schemes.

Abstract

Implementing general functions of operators is a powerful tool in quantum computation. It can be used as the basis for a variety of quantum algorithms including matrix inversion, real and imaginary-time evolution, and matrix powers. Quantum signal processing is the state of the art for this aim, assuming that the operator to be transformed is given as a block of a unitary matrix acting on an enlarged Hilbert space. Here we present an algorithm for Hermitian-operator function design from an oracle given by the unitary evolution with respect to that operator at a fixed time. Our algorithm implements a Fourier approximation of the target function based on the iteration of a basic sequence of single-qubit gates, for which we prove the expressibility. In addition, we present an efficient classical algorithm for calculating its parameters from the Fourier series coefficients. Our algorithm uses only one qubit ancilla regardless the degree of the approximating series. This contrasts with previous proposals, which required an ancillary register of size growing with the expansion degree. Our methods are compatible with Trotterised Hamiltonian simulations schemes and hybrid digital-analog approaches.