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

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, 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.

Figures (10)

• Figure 1: Quantum Circuit Diagram of HHL. Top Row: The "b register" where the right-hand side of $A\mathbf{p}=\mathbf{b}$ is encoded, and the final state, $\ket{x} \propto A^{-1}\mathbf{b}$ is found at the end. Middle Row: The "clock register" where the eigenvalues of $A$ are estimated and subsequently inverted. Bottom Row: The ancilla qubit, used to determine if the entangled b register and clock register collapsed to the solution state.
• Figure 2: HHL Solution to the 1D Poisson Equation, $\frac{d^2u}{dx^2}=f(x), f(x) =10x$, domain $[0,1]$, inhomogeneous boundary conditions: $u(0)=0, u(1) = 1$. 16 grid points using $n=4$ problem qubits, $n_c = 8$ clock qubits, and $R=150$ Trotter steps.
• Figure 3: a) Solution to the 2D Poisson Equation, $\nabla^2u=f(x,y), f(x,y) =4- 8H(x-0.5)$, domain $[0,1]^2$, inhomogeneous boundary conditions: $u(0, y) = 0.5, u(1, y) = \sin(y), u(x, 0) = (x - 0.5)(x - 1.0), u(x, 1) = 0.5(x - 1.0)$. a) Classical Benchmark. b) HHL Solution, 16$\times$16 grid using $n=8$ problem qubits, $n_c = 8$ clock qubits, and $R=150$ Trotter steps. c) ARE between Classical and HHL Solution.
• Figure 4: Average ARE of HHL Poisson Solver to the 1D Poisson Equation, $\frac{d^2u}{dx^2}=f(x), f(x) =10x$, domain $[0,1]$, inhomogenous boundary conditions: $u(0)=0, u(1) = 1$. 16 grid points using $n=4$ problem qubits, as a Function of $n_c$, the number of clock qubits.
• Figure 5: Comparison of 2D Lid-Driven Cavity flow with Full State Vector Extraction. $Re=100$, $16 \times 16$ grid, $n_c=8$ clock qubits, $R=150$ Trotter steps. Classical Benchmark: a) of Pressure. b) of Velocity. HHL Quantum-Classical Hybrid Solution c) of Pressure. d) of Velocity. ARE: e) of Pressure Gradient. f) of Velocity.