Quantum Neural Physics: Solving Partial Differential Equations on Quantum Simulators using Quantum Convolutional Neural Networks
Authors: Jucai Zhai, Muhammad Abdullah, Boyang Chen, Fazal Chaudry, Paul N. Smith, Claire E. Heaney, Yanghua Wang, Jiansheng Xiang, Christopher C. Pain
Organizations: Resource Geophysics Academy, Imperial College London, London, SW7 2AZ UK · Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ UK · Quantum AI Leadership (QUAIL) Ltd, London, WC2H 9JQ UK · Centre for AI-Physics Modelling, Imperial-X, White City Campus, Imperial College London, London, W12 7SL UK · Data Assimilation Lab., Data Science Institute, Imperial College London, London, SW7 2AZ UK
Neural Physics recasts local discretisations of partial differential equations (PDEs) as fixed convolutional operators, providing a physics-preserving alternative to data-driven surrogate modelling in scientific machine learning. However, existing realizations remain largely confined to classical AI hardware and do not directly connect to quantum structured operator design. To bridge this gap, we introduce a Quantum Neural Physics framework and develop a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). The proposed method maps analytically prescribed stencil operators to local quantum convolutional primitives and embeds them within a classical multilevel W-cycle architecture, combining the operator-centric view of scientific ML with the numerical rigor of multigrid solvers. Using amplitude encoding together with the Linear Combination of Unitaries (LCU) and the Quantum Fourier Transform (QFT), the resulting local quantum operators admit logarithmic-depth implementation, with circuit depth scaling as O(logK) for an encoded block of size K under the idealized parallel circuit model considered here. Numerical experiments on Poisson, transient diffusion, convection--diffusion, and incompressible Navier--Stokes problems demonstrate numerical consistency, stable multilevel behaviour, and workflow-level feasibility on noiseless simulators. Comparisons with representative quantum linear solver paradigms further show that the main strength of HQC-CNNMG lies in its balanced trade-off among local circuit depth, numerical robustness, and compatibility with PDE structure, rather than in fully quantum global inversion.
Figures & tables
Figure 1: Schematic diagram of the multigrid W-cycle iteration. The figure illustrates the recursive process from the fine grid Gh to the coarsest grid G4h . The operator actions correspond to the notation introduced in the text and together form a multiscale encoder–decoder-like computational structure reminiscent of U-Net.
Figure 2: Representative packed-block quantum convolution circuit for the case s=3 and B=4 , corresponding to the mapping 4×4→2×2 . The circuit uses 8 qubits in total (4 ancilla qubits and 4 data qubits), with a gate count of 99 and a circuit depth of 53. This example provides a concrete instantiation of the LCU–QFT construction and is consistent with the logarithmic-depth and polylogarithmic gate-complexity scaling of the underlying local operator design.
Figure 3: Operator-level verification for the linear system Ax=b . The first row corresponds to the 16×32 grid and the second row to the 24×48 grid. From left to right, the panels show the classical reference solution obtained by spsolve , the HQC-CNNMG solution, and the corresponding spatial distribution of the relative error.
Figure 4: Elliptic benchmark for the Poisson equation on a 24×40 grid. The panels show the source term, the classical reference solution, the HQC-CNNMG solution, and the associated relative- and absolute-error fields, together with the reconstructed potential distribution.
Figure 5: Transient diffusion benchmark on a 16×24 grid. The upper panels compare the classical reference solutions, HQC-CNNMG solutions, and absolute-error fields at representative time steps. The bottom panels report the temporal evolution of the L2 error and relative error under repeated implicit solves.
Figure 6: Field-level validation of the convection–diffusion benchmark on a 32×32 grid. The panels compare analytical and HQC-CNNMG solutions at representative time steps together with the corresponding relative-error fields.
Figure 7: Reduced-quantity validation of the convection–diffusion benchmark on a 32×32 grid. The panels compare the final-time solution profiles, the trajectory of the pulse centre, and the temporal attenuation of the peak amplitude, thereby assessing the preservation of key transport observables.
Figure 8: Flow past a square cylinder at Re=120 and t=900s . Left: vorticity field showing alternating wake shedding behind the obstacle. Right: velocity magnitude, velocity vectors, and streamlines, illustrating the unsteady recirculating wake structure consistent with a Kármán vortex street.
Algorithm Model
Space Complexity
Total Gate Count
Physical Depth
Preprocessing / Encoding Cost
Observable / Readout Cost
VQLS
O(logN)
O(poly(κ)log(1/ϵ)logN)
O(L)
O(logN)
O(1/ϵ2)
HHL
O(logN)+1anc+nclk
O(logN⋅s2κ2/ϵ)
O(s2κ2/ϵ)
O(logN)
O(1)
QSVT
O(logN)+O(logN)anc
O(κlog(1/ϵ)polylog(N))
O(κlog(1/ϵ))
O(logN)
O(1)
QUBO
O(Nlog(1/ϵ))
O(eN)
O(ecNlog(1/ϵ))
O(N2)
O(1)†
HQC-CNNMG (Block Encoding)
O(logK2)+1anc
O(K2NCBE(K,ϵloc))
O(DBE(K))
O(K2)
O(K2/ϵobs2)
HQC-CNNMG (LCU-QFT)
O(logK2)+4anc
O(K2NCLCU(K,ϵloc))
O(logK)
O(K2)
O(K2/ϵobs2)
Table 1: Paradigm-level leading-order resource scaling for representative quantum linear solver approaches applied to an N -dimensional discretized system. For HQC-CNNMG, K denotes the side length of the local convolution block or sliding window.
Solver Model
Reached Target Threshold?
Mean Relative L2 Error
Mean Iterations / Cycles
VQLS
Yes
0.009016
59.0
HHL
No
0.077856
1.0
QSVT
Yes
<10−6
1.0
QUBO
No
0.662328
1.0
HQC-CNNMG (Block Encoding)
Yes
0.000080
6.0
HQC-CNNMG (LCU-QFT)
Yes
0.000080
6.0
Table 2: Benchmark convergence statistics for the N=16 linear-system test, averaged over three independent runs. A maximum iteration cap is imposed on VQLS, while QUBO uses finite-bitwidth approximations for continuous variables together with heuristic sampling.
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.
Fabio Pereira dos Santos, Renato Portugal, Júlio de Castro Vargas Fernandes +1
Coordination of Mathematical and Computational Methods National Laboratory of Scientific Computing Av. Getúlio Vargas, 333 - Quitandinha, Petrópolis - RJ, 25651-075 · Center for Computation and Technology - Louisiana State University, 888 S. Stadium Dr., Baton Rouge, LA 70808, USA
We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.
Xiang Rao, Yuxuan Shen
1. School of Petroleum Engineering, Yangtze University, Wuhan 430100, China · 2. School of Computer Science, Yangtze University, Jingzhou 434023, China · 3. State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University), Wuhan 430100, China +2
Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning.
Chanyoung Kim, Myeonghwan Seong, Yujin Kim +2
KAIST · Department of Statistics and Data Science, Yonsei University · Department of Applied Statistics, Yonsei University +2