Quantum Algorithm for the Advection-Diffusion Equation by Direct Block Encoding of the Time-Marching Operator

TL;DR

This paper introduces a quantum algorithm for simulating multidimensional scalar transport problems using a direct block encoding of the time-marching operator, enabling efficient simulation without amplitude amplification.

Contribution

It presents a novel quantum algorithm that encodes the explicit time-marching operator directly, improving efficiency for multidimensional advection-diffusion simulations.

Findings

Successful state-vector simulations of 2D scalar transport in a vortex

The algorithm maintains linear dependence on simulation time

Supports theoretical efficiency claims with numerical results

Abstract

A quantum algorithm for simulating multidimensional scalar transport problems using a time-marching strategy is presented. A direct unitary block encoding of the explicit time-marching operator is constructed, resulting in the intrinsic success probability of the squared solution norm without the need for amplitude amplification, thereby retaining a linear dependence on the simulation time. The algorithm separates the explicit time-marching operator into an advection-like component and a corrective shift operator. The advection-like component is mapped to a Hamiltonian simulation and combined with the shift operator through the linear combination of unitaries algorithm. State-vector simulations of a scalar transported in a steady two-dimensional Taylor-Green vortex support the theoretical findings.