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Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural operators for general geometries · 0 citations (OpenAlex)

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Proposes NVQLS, a hybrid quantum-classical unsupervised spectral operator learner that combines a Legendre-Galerkin weak form, a phase-aware VQLS overlap loss, and a classical angle network mapping PDE inputs to circuit parameters.

Problem

Repeatedly solving parametric PDEs is costly; existing unsupervised operator learners (PI-DON, SCLON) scale poorly with dimension and basis size, and VQLS-based solvers are instance-specific and suffer from sign/phase ambiguity.

Approach

Combine VQLS with a Legendre-Galerkin weak formulation and an angle network that maps each forcing/coefficient instance to variational circuit parameters, using a real-part overlap loss that removes sign ambiguity and Pauli-grouped measurements to cut circuit evaluations.

Key results

• Mean relative L2 and Linf errors on 1D and 2D benchmarks are in the 1e-2 to 1e-3 range; some cases (1D reaction-diffusion, joint Helmholtz) reach about 0.1%.
• On the 1D wave equation, even NVQLS's worst-case test sample (error ~1e-3) outperforms the PI-DON prediction.
• Neural embedding with 2 layers and 1,304 parameters reportedly achieves convergence and generalization that purely quantum baselines with 100 layers (~1,200 parameters) do not reach at D=50.

Surprising findings

• Purely quantum sequential-embedding and data re-uploading baselines fail to train well as the number of instances grows, while the hybrid neural embedding does not.

Limitations

• Complexity advantages rely on assumed polylogarithmic-depth circuits and efficient state preparation, and the worst-case f-embedding depth can be O(N^d).
• Experiments are simulated noiseless PennyLane/JAX runs on small problems (1D and 2D, small N), with no hardware results.
• Phase-aware loss derivation depends on assumptions about real-valued solutions and state normalization; the quadratic loss variant is only motivated heuristically.

Experimental setup

Datasets: forcing functions generated for 1D and 2D PDEs (synthetic)

Benchmarks: 1D reaction-diffusion 1D Helmholtz 1D convection-diffusion 1D wave equation 2D Helmholtz

Metrics: relative L2 error L-infinity error MAE number of Pauli measurements

Baselines: PI-DON SCLON VQLS sequential embedding data re-uploading

Builds on: VQLS SCLON PI-DON Legendre-Galerkin method

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