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

2026-03-25

AI Summary

This work addresses the computational and memory bottlenecks of classical mesh-based methods in solving partial differential equations (PDEs) with billions of degrees of freedom. The authors propose a quantum neural physics framework that, for the first time, maps discrete differential operators to parameter-free quantum convolutional kernels of depth \(O(\log K)\), which are embedded within a classical U-Net multigrid solver to form a hybrid quantum-classical PDE solver. Leveraging amplitude encoding, linear combinations of unitaries, the quantum Fourier transform, and quantum convolutional networks, the method achieves exponential memory compression and potential computational speedup. Numerical experiments on a quantum simulator demonstrate accurate solutions to the Poisson equation, diffusion equation, convection-diffusion equation, and incompressible Navier–Stokes equations, with results in close agreement with classical approaches.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• Partial Differential Equations
• Computational Bottleneck
• High-Dimensional Simulation
• Scientific Computing
• Quantum Simulation

Innovation

• Quantum Convolutional Neural Networks
• Hybrid Quantum-Classical Solver
• Amplitude Encoding
• Linear Combination of Unitaries
• Multigrid Methods