Quantum Hamiltonian simulation of linearised Euler equations in complex geometries

TL;DR

This paper extends quantum Hamiltonian simulation techniques to complex geometries in fluid dynamics, enabling more realistic PDE modeling with efficient quantum circuits and boundary condition handling.

Contribution

It introduces methods for incorporating complex boundary conditions into quantum PDE simulations and constructs explicit circuits for linearized Euler equations with obstacles.

Findings

Quantum circuits successfully simulate linearized Euler equations with complex boundaries.

Boundary conditions do not increase Trotter error or circuit complexity.

Quantum solutions show comparable accuracy to classical finite difference methods.

Abstract

Quantum computing promises exponential improvements in solving large systems of partial differential equations (PDE), which forms a bottleneck in high-resolution computational fluid dynamics (CFD) simulations, in, among others, aerospace applications and weather forecasting. One approach is via mapping classical CFD problems to a quantum Hamiltonian evolution, for which recently an explicit quantum circuit construction has been shown in simple cases, allowing proof-of-concept execution on quantum processors. Here we extended this method to more complex and practically relevant cases. We first demonstrate how boundary conditions corresponding to arbitrary complex-shaped obstacles can be introduced in the quantum representations of elementary difference operators used to implement the PDE. We provide explicit and efficient circuit constructions, and show they neither increase the Trotter error, nor asymptotic gate complexity with respect to the free space equation. Using these methods we then derive quantum circuits for simulating the linearized Euler equations in a presence of a background fluid flow and obstacles. We illustrate our results by simulating the obtained quantum circuits for a number of boundary conditions, and compare the errors of the quantum solution to classical finite difference methods.