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

TL;DR

This paper introduces a hybrid quantum-classical algorithm to solve the incompressible Navier-Stokes equations, demonstrating feasibility on current noisy quantum hardware and proposing techniques to optimize quantum resource use.

Contribution

It presents a novel hybrid variational algorithm for fluid dynamics that combines classical nonlinear computations with quantum pressure solving, including a new quantum readout method and resource optimization strategies.

Findings

High-fidelity results achieved on noisy quantum hardware.

Multigrid preconditioning reduces quantum resource requirements.

HTree readout technique is effective for real-valued problems.

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.