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  • Letter
  • Open Access

Quantum algorithm for the advection-diffusion equation by direct block encoding of the time-marching operator

Paul Over1, Sergio Bengoechea1, Peter Brearley2, Sylvain Laizet2, and Thomas Rung1

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Phys. Rev. A 112, L010401 – Published 1 July, 2025

DOI: https://doi.org/10.1103/d8hb-fv93

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.

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Phys. Rev. A 112, L010401– Published 1 July, 2025
  • Received 11 October 2024
  • Accepted 4 June 2025
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