An HHL-Based Quantum-Classical Solver for the Incompressible Navier-Stokes Equations with Approximate QST

TL;DR

This paper introduces a hybrid quantum-classical solver for incompressible Navier-Stokes equations using HHL and a novel quantum state tomography method, demonstrating accurate fluid flow simulations and addressing measurement challenges.

Contribution

It couples HHL quantum linear system algorithms with a new QST approach to enable practical quantum CFD simulations, overcoming measurement bottlenecks.

Findings

Successfully simulated lid-driven cavity flow and Taylor-Green vortex with quantum methods.

Validated hybrid quantum-classical solver against classical numerical methods.

Demonstrated potential for quantum algorithms to accelerate CFD computations.

Abstract

In computational fluid dynamics (CFD), the numerical integration of the Navier-Stokes equations is frequently constrained by the Poisson equation to determine the pressure. Discretization of this equation often results in the need to solve a system of linear algebraic equations. This step typically represents the primary computational bottleneck. Quantum linear system algorithms such as Harrow-Hassidim-Lloyd (HHL) offer the potential for exponential speedups for solving sparse linear systems, such as those that arise from the discretized Poisson equation. In this work, we successfully couple HHL to a discretized formulation of the incompressible Navier-Stokes equations and demonstrate both accurate lid-driven cavity flow simulations as a fully integrated benchmark problem, and accurate flow of the Taylor-Green vortex. To address the readout limitation, we utilize a recent novel quantum state tomography (QST) approach based on Chebyshev polynomials and Quantum Amplitude Estimation (QAE), which enables approximate statevector extraction without full state reconstruction. Together, these results clarify the algorithmic structure required for quantum CFD, explicitly confront the measurement bottleneck, and establish benchmark problems for future quantum fluid simulations. We implement the solver using IBM's Qiskit framework and validate the hybrid quantum-classical simulation against standard classical numerical methods. Our results demonstrate that the hybrid solver successfully captures the global vortex dynamics of the lid-driven cavity problem and the Taylor-Green vortex, offering a robust pathway for integrating quantum subroutines into more practical higher-Reynolds number CFD workflows.