Quantum Hamiltonian simulation of linearised Euler equations in complex geometries

Abstract

Quantum computing promises exponential improvements in solving large systems of partial differential equations (PDE), which forms a bottleneck in high-resolution simulations of, among others, computational fluid dynamics (CFD) in aerospace applications and weather forecasting. One approach is via mapping classical PDE 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 arbitrary complex-shaped geometrical obstacles with Dirichlet, Neumann or mixed boundary conditions can be introduced in the quantum representations of elementary difference operators used to implement the PDE, either directly or using Linear Combination of Unitaries (LCU). We provide their explicit and efficient circuit constructions, and analyze the Trotter errors and asymptotic gate complexities, which in the Dirichlet case do not grow compared to the free space equation. Using these methods we then derive quantum circuits for the linearized Euler equations in the presence of a background fluid flow and obstacles, both in the conservative and non-conservative regimes. We illustrate our results by simulating the obtained quantum circuits for different boundary conditions and geometries, comparing their error to a classical finite difference scheme.