Incompressible Navier-Stokes solve on noisy quantum hardware via a hybrid quantum-classical scheme

TL;DR

This work presents a hybrid quantum–classical scheme for solving the incompressible Navier–Stokes equations on noisy intermediate-scale quantum hardware by combining a classical pressure-projection step with a variational quantum solver for the pressure Poisson equation. A preconditioned variational approach, notably algebraic multigrid (AMG), improves the trainability and convergence of the quantum solver, while Hadamard-tree (HTree) readout enables efficient extraction of real-valued pressure states. The method is demonstrated on a 2D lid-driven cavity benchmark, with simulations on noise-free and IBM Q hardware showing high fidelity under modest noise and highlighting current hardware limitations for explicit time-stepping CFD on NISQ devices. Resource comparisons indicate VQE is the most practical quantum solver for near-term implementations, whereas VQLS and HHL face prohibitive requirements on NISQ hardware for this problem. This work advances practical quantum CFD by integrating preconditioning and real-valued readout to mitigate noise in a realistic hybrid setting, and it provides a path toward scalable quantum-assisted fluid dynamics on near-term devices, with code available under MIT license.

Abstract

Partial differential equation solvers are required to solve the Navier-Stokes equations for fluid flow. Recently, algorithms have been proposed to simulate fluid dynamics on quantum computers. Fault-tolerant quantum devices might enable exponential speedups over algorithms on classical computers. However, current and foreseeable quantum hardware introduce noise into computations, requiring algorithms that make judicious use of quantum resources: shallower circuit depths and fewer qubits. Under these restrictions, variational algorithms are more appropriate and robust. This work presents a hybrid quantum-classical algorithm for the incompressible Navier--Stokes equations. A classical device performs nonlinear computations, and a quantum one uses a variational solver for the pressure Poisson equation. A lid-driven cavity problem benchmarks the method. We verify the algorithm via noise-free simulation and test it on noisy IBM superconducting quantum hardware. Results show that high-fidelity results can be achieved via this approach, even on current quantum devices. Multigrid preconditioning of the Poisson problem helps avoid local minima and reduces resource requirements for the quantum device. A quantum state readout technique called HTree is used for the first time on a physical problem. Htree is appropriate for real-valued problems and achieves linear complexity in the qubit count, making the Navier-Stokes solve further tractable on current quantum devices. We compare the quantum resources required for near-term and fault-tolerant solvers to determine quantum hardware requirements for fluid simulations with complexity improvements.

Figures (10)

• Figure 1: We show a schematic of the hybrid quantum--classical scheme for solving incompressible flow problems. (a) Velocity contour of the 2D lid-driven cavity flow at $\hbox{Re}\xspace = 1000$. (b) A staggered grid allays the well-known checkerboard pressure problem. The pressure is stored at the cell center, and the velocities are stored at the cell interfaces. (c) Calculate intermediate velocities and proceed to the classical computer's next fractional time step. (d) Solve the Poisson equation with proper preconditioning on a quantum computer and obtain velocity corrections. We iterate until the convergence criterion reaches the threshold $\epsilon$.
• Figure 2: Quantum circuit structure for the (a) VQE and (b) VQLS algorithms.
• Figure 3: Comparison of the (a) original optimization landscape and (b) preconditioned landscape. The preconditioned landscape (b) has fewer local minima and thus exhibits better trainability. The global minima are located at the center $(0,0)$ in this illustration. $\text{PC}_j$ is the $j$-th principal component.
• Figure 4: Lid-driven cavity flow at $\hbox{Re}\xspace = 100$ over a $100 \times 100$ grid. (a) Velocity magnitude $\|\bm{u}\|= \sqrt{u^2+v^2}$ at $t = 10$. (b) Benchmark results and a comparison against a reference solution. The horizontal curve is $y$-direction velocity $v$ along the horizontal line through the geometric center of the cavity at $y=0.5$. The vertical curve compares the $x$-direction velocity $u$ along the vertical line through the geometric center of the cavity at $x=0.5$ with the reference values of ghia1982high. The convergence criterion is $r_{\text{RMS}} \leq 10^{-5}$.
• Figure 5: Preconditioning helps VQE converge faster to solve the linear system on a $5 \times 5$ mesh. The data are averaged from five VQE runs with random initialization. The PQC is chosen to be the real-amplitude ansatz with five repetitions. The black line shows results without preconditioning. ILU is an incomplete LU factorization and AMG is algebraic multigrid. The number of iterations reported here could differ from other literature recordings of the number of function evaluations. For the L-BFGS-B optimizer, each iteration requires about $30$ evaluations to approximate gradient information.