Quantum Neural Physics: Solving Partial Differential Equations on Quantum Simulators using Quantum Convolutional Neural Networks

In scientific computing, the formulation of numerical discretisations of partial differential equations (PDEs) as untrained convolutional layers within Convolutional Neural Networks (CNNs), referred to by some as Neural Physics, has demonstrated good efficiency for executing physics-based solvers on GPUs. However, classical grid-based methods still face computational bottlenecks when solving problems involving billions of degrees of freedom. To address this challenge, this paper proposes a novel framework called 'Quantum Neural Physics' and develops a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). This approach maps analytically-determined stencils of discretised differential operators into parameter-free or untrained quantum convolutional kernels. By leveraging amplitude encoding, the Linear Combination of Unitaries technique and the Quantum Fourier Transform, the resulting quantum convolutional operators can be implemented using quantum circuits with a circuit depth that scales as O(log K), where K denotes the size of the encoded input block. These quantum operators are embedded into a classical W-Cycle multigrid using a U-Net. This design enables seamless integration of quantum operators within a hierarchical solver whilst retaining the robustness and convergence properties of classical multigrid methods. The proposed Quantum Neural Physics solver is validated on a quantum simulator for the Poisson equation, diffusion equation, convection-diffusion equation and incompressible Navier-Stokes equations. The solutions of the HQC-CNNMG are in close agreement with those from traditional solution methods. This work establishes a mapping from discretised physical equations to logarithmic-scale quantum circuits, providing a new and exploratory path to exponential memory compression and computational acceleration for PDE solvers on future fault-tolerant quantum computers.