Quantum algorithm for solving differential equations using SLAC derivatives

TL;DR

This paper develops quantum algorithms for differential equations using SLAC derivatives, employing block-encodings, wavelet transforms, and preconditioning to improve efficiency and condition numbers.

Contribution

It introduces efficient quantum block-encodings for SLAC derivatives, multi-scale wavelet representations, and preconditioning techniques for solving PDEs on quantum computers.

Findings

High success probability and low gate cost in state preparation

Efficient multi-scale representations via Shannon wavelet transforms

Reduced condition number enabling faster quantum linear solvers

Abstract

We present the construction of efficient linear-combination-of-unitaries (LCU)-based block-encodings for the first-order derivative and Laplacian operators in the SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to efficiently prepare the dense LCU amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to efficiently obtain multi-scale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multi-scale wavelet basis to a small constant. This approach enables the solution of partial differential equations with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm (QLSA). Throughout this work, we analyse the computational complexity and error scaling of each implementation.