A quantum advection-diffusion solver using the quantum singular value transform

TL;DR

This paper introduces a quantum algorithm for simulating the advection-diffusion equation that leverages high-order finite-difference methods and quantum singular value transform to improve efficiency and reduce resource requirements.

Contribution

It develops a novel quantum algorithm that combines block encodings and the quantum singular value transform for efficient advection-diffusion simulation, demonstrating reduced complexity.

Findings

Higher order methods decrease the number of gates and qubits needed.

Numerical simulations confirm the theoretical efficiency gains.

The approach is applicable to 1D and 2D benchmark problems.

Abstract

We present a quantum algorithm for the simulation of the linear advection-diffusion equation based on block encodings of high order finite-difference operators and the quantum singular value transform. Our complexity analysis shows that the higher order methods significantly reduce the number of gates and qubits required to reach a given accuracy. The theoretical results are supported by numerical simulations of one- and two-dimensional benchmarks.