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.
@article{arxiv.2410.07909,
title = {Quantum Algorithm for the Advection-Diffusion Equation by Direct Block Encoding of the Time-Marching Operator},
author = {Paul Over and Sergio Bengoechea and Peter Brearley and Sylvain Laizet and Thomas Rung},
journal= {arXiv preprint arXiv:2410.07909},
year = {2025}
}