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arXiv:2510.17978

Quantum Hamiltonian simulation of linearised Euler equations in complex geometries

Vladyslav Bohun, Andrij Romanovych Kuzmak, Maciej Koch-Janusz

Published Oct 20, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2510.17978

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We construct explicit quantum circuits for the velocity-stress formulation of elastic wave equations in homogeneous media using Schrödingerisation. After rewriting the equations in symmetric matrix form(SMF), we use Fourier spectral discretization for source-driven periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundary conditions. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs -with unitary evolution- using the warped phase transformation that maps the equations in one higher dimension. For periodic problems, we develop circuits by combining controlled Fourier-mode rotations. For physical boundaries, interleaved encoding swaps the spatial and field-component registers, enabling a more efficient circuit construction using shift operators. We analyze pppth-order Trotter-Suzuki formulas and derive splitting-error and CNOT-gate estimates, under the assumptions of smooth auxiliary initialization and solution-state recovery. Comparison with classical partitioned splitting methods identifies a polynomial improvement in spatial-resolution dependence for the quantum state-evolution subroutine. Noiseless circuit simulations of a forced periodic problem and an impedance problem illustrate the constructions and quantify their errors against exact solutions and classical reference computations.
Solving partial differential equations for extremely large-scale systems within a feasible computation time serves in accelerating engineering developments. Quantum computing algorithms, particularly the Hamiltonian simulations, present a potential and promising approach to achieve this purpose. Actually, there are several oracle-based Hamiltonian simulations with potential quantum speedup, but their detailed implementations and accordingly the detailed computational complexities are all unclear. This paper presents a method that enables us to explicitly implement the quantum circuit for Hamiltonian simulation; the key technique is the explicit gate construction of differential operators contained in the target partial differential equation discretized by the finite difference method. Moreover, we show that the space and time complexities of the constructed circuit are exponentially smaller than those of conventional classical algorithms. We also provide numerical experiments and an experiment on a real device for the wave equation to demonstrate the validity of our proposed method. Published by the American Physical Society 2024