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

Abstract

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.

Figures (8)

• Figure 1: Schematic diagram of the multigrid W-Cycle iteration. The figure illustrates the recursive process from the fine grid $\mathcal{G}_h$ to the coarsest grid $\mathcal{G}_{4h}$. The operator operations correspond one-to-one with the symbols defined in the text and structurally present multi-scale Encoder-Decoder characteristics similar to U-Net.
• Figure 2: Quantum convolution circuit for $K\times K \to (K-2)\times(K-2)$ (taking $K=4$ as an example). This circuit uses 8 qubits (4 ancilla + 4 data), with a gate count of 99 and a circuit depth of 53, maintaining the circuit depth at the $O(\log K)$ level.
• Figure 3: Comparison of linear system solving results. The first row displays results for a grid size of $16 \times 32$ and the second row for $24 \times 48$. From left to right, the figures show: the exact solution provided by the SciPy classical direct solver, the numerical solution obtained by the HQC-CNNMG solver and the spatial distribution of the relative error between the two.
• Figure 4: Results of the Poisson equation solution. The first row shows the source term, the exact solution and the numerical solution of the Poisson equation. The second row displays the relative error, absolute error of the solution and the potential field distribution obtained by the solver. The grid size is $24 \times 40$.
• Figure 5: Comparison of diffusion equation solutions ($16 \times 24$ grid). Top four rows: Comparison of exact solutions, numerical solutions and absolute errors at different time steps; Bottom row: Evolution curves of $L^2$ error and relative error over time.