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Quantum SEDONet: Spectrally-Embedded Quantum Deep Operator Networks for Partial Differential Equations

Muhammad Abid, Arth Sojitra, Bipin Tiwari, Omer San

arXiv:2608.27626v1quant-phcs.LG

TL;DR

Quantum DeepONet’s trunk receives coordinates with limited spectral structure, leaving nonlinearities to reconstruct oscillations and making many-query PDE workflows costly. Quantum SEDONet adds boundary-matched Fourier or Chebyshev trunk embeddings, improving accuracy across four benchmarks without changing qubit count or circuit depth.

  • Problem

    Quantum operator-learning workflows need efficient repeated PDE solves, while raw trunk coordinates force networks to reconstruct oscillatory solution features through nonlinearities.

  • Method

    Quantum SEDONet assigns Fourier expansions to periodic coordinates and Chebyshev expansions to bounded, non-periodic coordinates, while preserving the Quantum DeepONet architecture.

  • Results

    36% to 54.1% reduction in mean relative L2 error across four benchmarks, with unchanged qubit count and circuit depth and only a few-percent parameter overhead.

  • Takeaways & Limitations

    Matching spectral bases to coordinate boundary conditions targets diagnosed high-frequency errors while retaining the baseline quantum resource requirements.

  • Takeaways & Limitations

    Results use ideal noiseless infinite-shot evaluation and a single training seed per arm, leaving realistic-noise and variability assessments for future work.

Abstract

from arXiv · show

Quantum DeepONet accelerates neural-operator inference by evaluating an orthogonally parameterized network on a quantum computer, reproducing in ideal simulation the accuracy of its classical counterpart at asymptotically lower inference cost. Its trunk network, however, receives query coordinates with limited spectral structure, requiring the network to learn oscillatory features through its nonlinearities. We propose Quantum SEDONet (Spectral-Embedded Deep Operator Network), which assigns each trunk coordinate a spectral basis according to its boundary condition: Fourier features for periodic coordinates and Chebyshev features for bounded, non-periodic coordinates. The basis is selected per coordinate rather than per problem, allowing both representations within a single problem. Under unary amplitude encoding, the embedding incurs no additional qubits or circuit depth when its dimension remains within the network width, while increasing the parameter count by only a few percent. Across four benchmarks, Quantum SEDONet reduces the mean relative L2 error by 54.1% for the antiderivative, 49.6% for advection, 36.0% for Burgers, and 36.2% for a mixed-boundary channel Poisson problem. Quantum and classical evaluation paths agree to within 10^-8 throughout. The channel Poisson problem simultaneously uses Fourier features in the periodic direction and Chebyshev features in the bounded direction, demonstrating coordinate-wise boundary-matched spectral embedding without additional quantum-resource cost.

1 Introduction

The paper targets the high per-solve cost of PDE computation and the limited spectral structure supplied to Quantum DeepONet trunks. Quantum SEDONet adds boundary-matched spectral coordinates while preserving the quantum resource profile when embeddings fit the network width.

  • Motivation: PDE solvers incur their computational cost separately for every changed boundary condition, forcing, parameter, or geometry.This makes many-query workflows such as uncertainty quantification, design optimization, and digital twins impractical.
  • Related approaches: Neural operators learn mappings between function spaces and provide mesh-free, resolution-flexible inference after training.DeepONet implements this mapping with branch and trunk subnetworks.
  • Quantum DeepONet: Quantum DeepONet replaces dense branch and trunk layers with orthogonal RBS-gate layers, training classically and transferring parameters for quantum inference.This division of labor addresses trainability concerns associated with variational quantum models.
  • Motivation: Raw trunk coordinates force nonlinearities to synthesize oscillations, while coordinate networks preferentially learn low frequencies.The resulting approximation error concentrates where the solution varies rapidly.
  • Quantum SEDONet: Quantum SEDONet assigns Fourier expansions to periodic coordinates and Chebyshev expansions to bounded, non-periodic coordinates on a per-coordinate basis.This imports classical spectral basis selection into the trunk input map.
  • Resource implications: An embedding no larger than the network width adds no qubits or circuit depth under unary encoding, with only a few-percent parameter increase.The network width already dominates the trunk input dimension in the stated regime.

2 Related Work

Related work spans neural operators, spectral coordinate embeddings, boundary-aware scientific machine learning, and quantum approaches to PDEs and operator learning. Quantum SEDONet combines these lines through boundary-driven spectral input selection for a unary orthogonal architecture.

  • Neural operators and reduced-order modeling: Neural-operator research extends DeepONet through multiple input functions, multifidelity data, physics-informed training, gradient balancing, and learned-basis analyses.The paper identifies trunk input representation as a way to control which bases are reachable.
  • Spectral methods and embeddings: Fourier features accelerate high-frequency learning by mapping low-dimensional coordinates into higher-dimensional periodic feature spaces.The paper uses deterministic, low-order Fourier terms rather than random features.
  • Spectral methods and embeddings: Fourier series suit periodic problems, whereas Chebyshev polynomials provide a global basis for bounded, non-periodic and endpoint-sensitive structure.Quantum SEDONet transfers this basis criterion to trunk features instead of discretization.
  • Boundary-aware architectures: Boundary information can be encoded architecturally through hard constraints or Fourier substitutions rather than only through the loss.The paper states the corresponding rule for both periodic and bounded coordinates.
  • Quantum computing: Quantum PDE algorithms address solution procedures, while Quantum DeepONet and related models accelerate operator inference through learned quantum-compatible representations.The distinction separates direct quantum PDE solvers from the present surrogate-inference approach.
  • Quantum computing: In unary orthogonal layers, register width depends on the larger input or output dimension, so richer inputs remain resource-free until exceeding network width.This hardware property motivates spectral trunk enrichment.

3 Methods

The methods section develops the quantum foundation of Quantum SEDONet, from single-layer circuits through quantum orthogonal neural networks and Quantum DeepONet assembly.

  • Methods: The quantum foundation covers RBS gates, unary data loading, pyramidal orthogonal transformation, and tomography returning a classical vector.These components are assembled into QOrthoNN and then Quantum DeepONet.

3.1 Quantum methods for network layers

Quantum network layers load vectors into unary states, apply orthogonal transformations with RBS pyramids, and recover classical outputs through tomography. The design uses sparse unary structure and stable angle construction to control resources and numerical error.

  • Single quantum layer: A quantum layer loads a classical vector, applies an orthogonal matrix through a pyramidal circuit, and returns the result by tomography.Bias addition and nonlinear activation remain classical.
  • RBS gates: RBS gates rotate amplitudes within the single-excitation subspace while preserving the unary subspace.This lets the circuit represent vector transformations without using the full Hilbert space.
  • Data loading: Unary loading places each normalized input component in the amplitude of its corresponding unary basis state.When dimensions differ, the register uses max(m, n) qubits.
  • Orthogonal transformation: A pyramidal sequence of RBS gates realizes products of Givens rotations and maps the loaded amplitudes to the components of Wx.Excitation conservation keeps the state in the sparse unary subspace.
  • Tomography: Tomography recovers output magnitudes from probabilities and signs through interference with a known reference amplitude.An ancillary qubit supports the sign-recovery procedure.
  • Tomography: The tomography extraction costs O(n/δ2) measurements and has estimation error independent of register size r.For m < n, only the bottom m unary states are read.
  • Numerical stability: The released loader-angle procedure fails on sparse ReLU activations, where four to seven of ten width-10 components vanish and accumulated products diverge.This limitation motivates the cumulative-tail-norm reformulation.

3.2 Quantum DeepONet

Quantum DeepONet replaces dense branch and trunk layers with orthogonal quantum layers while preserving a classical training path. The branch encodes the input function, the trunk encodes query coordinates, and their inner product produces the operator evaluation.

  • DeepONet maps an input function v to a PDE solution through branch and trunk subnetworks whose outputs are combined by an inner product.
  • Quantum DeepONet replaces both multilayer perceptrons with QOrthoNN stacks, applying biases and nonlinearities classically between quantum layers.
  • The orthogonal quantum layers retain the learned operator in the ideal regime while providing asymptotically lower inference cost per layer.

3.3 Quantum SEDONet

Quantum SEDONet preserves Quantum DeepONet’s architecture but replaces raw trunk coordinates with boundary-matched spectral features. Fourier features serve periodic coordinates, while Chebyshev features serve bounded non-periodic coordinates without increasing quantum resources when the embedding fits the network width.

  • Quantum SEDONet modifies only how query coordinates enter the trunk, addressing the need to synthesize oscillations from raw linear inputs.The raw-coordinate trunk is biased toward low frequencies, concentrating approximation error where solutions vary fastest.
  • Periodic coordinates receive Fourier features, while bounded non-periodic coordinates receive Chebyshev features before the first trunk quantum layer.The embedding is applied as classical preprocessing to the trunk input.
  • Omitting the constant Chebyshev polynomial avoids a degenerate feature column, while the affine first polynomial retains the raw coordinate as a special case.This makes the embedding capable of recovering baseline behavior when that is optimal.
  • The architecture combines the branch and embedded trunk outputs through the same inner-product operator-evaluation structure as Quantum DeepONet.The spectral embedding is the single added block before the trunk quantum layers.
  • When the feature dimension is at most W −1, the embedding leaves qubit count and circuit depth unchanged and adds only a few percent to the parameter count.The first trunk layer still requires 1 + W qubits, and later width-to-width layers are unchanged.

3.4 Training quantum SEDONet

Quantum SEDONet is trained classically using the orthogonal parameterization, with the trained angles later transferred to quantum inference. The embedding is precomputed for the fixed trunk grid, and baseline comparisons hold all settings except the trunk input map fixed.

  • The pyramidal circuit matches a classical orthogonal neural network mathematically, enabling classical backpropagation followed by transfer of trained angles to quantum evaluation.
  • Baseline and Quantum SEDONet arms differ only in the trunk input map, while width, depth, optimizer settings, iteration count, and training data remain fixed.This isolates the spectral embedding as the experimental difference.
  • The training algorithm precomputes the boundary-matched embedding, checks dim ϕ ≤W −1, and caches feature values on the trunk grid.The dimension check guarantees the claimed unchanged qubit count.

4 Results

Across four operator benchmarks, Quantum SEDONet evaluates boundary-matched spectral embeddings against otherwise matched Quantum DeepONet baselines. Function-approximation tests validate classical–quantum agreement, while operator results show substantial error reductions, including a 54.1% reduction for the antiderivative.

  • 4 Results: The four operator benchmarks cover Chebyshev-only, mixed periodic–non-periodic, and newly introduced channel Poisson settings, exercising the full basis-selection rule.
  • 4.2 Function approximation: Classical OrthoNN and ideal quantum QOrthoNN predictions agree to 7.8 × 10−9 for the Runge function and 1.0 × 10−6 for the sinusoidal sum.The corresponding relative L2 errors are 0.136% and 3.308%.
  • 4.3 Antiderivative operator: The antiderivative embedding preserves the 11-qubit trunk register while increasing trunk pyramid angles from 17 to 45.
  • 4.3 Antiderivative operator: 54.1%: the antiderivative mean relative L2 error decreases from 1.159% to 0.532%, the largest reduction among the four benchmarks.The paired comparison favors Quantum SEDONet on 99 of 100 test functions.
  • 4.3 Antiderivative operator: The antiderivative baseline’s worst case exhibits ReLU-induced piecewise-linear kinks, whereas Chebyshev features track the reference almost exactly.Its roughly 20-times-smaller amplitude inflates the relative error, despite small absolute discrepancies.

4.4 Advection Equation

The advection benchmark tests operator learning for periodic transport, comparing Quantum DeepONet with Quantum SEDONet using identical training data. Spectral embedding reduces error by representing the periodic spatial coordinate spectrally and resolving temporal structure.

  • The operator maps sampled initial conditions to solutions at later times for one-dimensional advection with periodic spatial boundaries.
  • Fourier features represent the periodic spatial coordinate, while Chebyshev features supply temporal resolution absent from the raw coordinate.For pure translation, the solution acts on spatial Fourier modes as phase rotations.
  • A representative prediction shows the baseline error along diagonal transport characteristics, while spectral embedding removes nearly all of that structure.
  • 49.6%: mean relative L2 error falls from 3.463% to 1.744% across 200 advection test functions.
  • Quantum SEDONet is more accurate on 199 of 200 paired test functions, with a paired t-statistic of 23.3.

4.5 Burgers Equation

The Burgers benchmark evaluates trunk embeddings for a nonlinear periodic PDE whose viscosity smooths most of the solution while leaving sharp structure near the initial time. Quantum SEDONet reduces aggregate prediction error relative to Quantum DeepONet.

  • The operator maps initial conditions to later-time solutions of one-dimensional Burgers equation with periodic spatial boundaries and viscosity ν = 0.05.
  • Chebyshev temporal features substantially suppress the baseline error band at t = 0, where sharp initial-condition structure is hardest to resolve.
  • 36.0%: mean relative L2 error falls from 2.313% to 1.481% across 100 Burgers test functions.
  • Quantum SEDONet improves on the baseline for 93 of 100 functions, with a paired t-statistic of 8.1.
  • The improvement is smaller than for advection because viscosity confines sharp structure near t = 0, whereas advection transports fine detail across space-time.

4.6 Channel Poisson Equation

The channel Poisson benchmark applies Fourier features to the periodic x coordinate and Chebyshev features to the bounded wall-normal y coordinate. Quantum SEDONet substantially reduces error relative to the baseline, including across the aggregate test set.

  • Boundary-matched embedding: Fourier features encode periodic x, while Chebyshev features encode the bounded Dirichlet y coordinate in the mixed-boundary problem.The source fields are periodic in x and vanish at the walls; reference solutions satisfy the Dirichlet condition to machine precision.
  • Prediction comparison: The representative prediction comparison shows substantially suppressed error where the solution has its sharpest wall-normal variation.The baseline’s structured error is attributed to poor resolution of the raw (x, y) trunk input, while the spectral terms match the periodic structure.
  • Aggregate results: 36.2%: Quantum SEDONet reduces mean relative L2 error from 5.889% to 3.760% on channel Poisson.The comparison is aggregated over the test set.
  • Aggregate results: 98%: Quantum SEDONet improves on the baseline for this benchmark’s test functions, with paired t = 26.8.Figure 10 reports mean relative L2 error with standard-deviation bars.
  • Interpretation: Channel Poisson is the first benchmark where both spectral bases are dictated by spatial boundary conditions within one problem.This demonstrates coordinate-wise selection rather than choosing one representation for the entire problem.

5 Summary and Conclusions

Quantum SEDONet modifies only the classical trunk input map, assigning boundary-matched spectral features while preserving the Quantum DeepONet architecture. Across four PDE regimes it improves accuracy with unchanged qubit count and circuit depth, though the evidence is limited to ideal noiseless simulations and smooth solutions.

  • Method: Quantum SEDONet assigns Fourier expansions to periodic coordinates and Chebyshev expansions to bounded, non-periodic coordinates without changing the quantum architecture or tomography procedure.The embedding is applied before the first quantum layer.
  • Accuracy: The embedding improves on the baseline for the overwhelming majority of identical test inputs on every benchmark under paired comparison.The antiderivative case is better on 99 of 100 test functions.
  • Resources: Qubit count and circuit depth remain unchanged on every benchmark, while the sole overhead is a few percent more trainable parameters.The additional parameters come from RBS angles in the first trunk layer; the antiderivative trunk register remains 11 qubits.
  • Mechanism: The improvement is linked to replacing raw affine coordinates with spectral structure where solution variation is fastest.The reported failure modes include transport-characteristic errors, the Burgers initial-time boundary layer, sharp wall-normal variation, and piecewise-linear antiderivative kinks.

6 Future Work

The paper identifies realistic-noise validation, multi-seed evaluation, adaptive spectral-degree selection, broader boundary and geometry coverage, and tests on nonsmooth solutions as future priorities.

  • Validation: Realistic hardware validation should repeat the comparison under depolarizing and device-calibrated noise, finite shots, and unary post-selection.The current results use ideal, noiseless, infinite-shot evaluation; identical baseline and proposed circuit resources may not guarantee identical noisy accuracy because RBS angles differ.
  • Validation: A multi-seed study reporting per-arm means and standard deviations is the primary outstanding validation step.The current paired statistics measure variability across test inputs, not across training initializations.
  • Method extensions: Adaptive Fourier and Chebyshev degree selection could replace the fixed resource-driven rule, especially when solution smoothness differs across coordinates.The current rule expands each coordinate until the embedding fills the network width, whereas higher-dimensional and multiphysics problems make width allocation more competitive.
  • Method extensions: Future extensions include Neumann conditions, semi-infinite or unbounded domains, and geometries that do not factor coordinate-wise.Laguerre and Hermite expansions are identified as classical choices for semi-infinite and unbounded domains.
  • Method extensions: Benchmarks with genuine interior discontinuities, such as shocks or material interfaces, are needed beyond the current smooth or strongly smoothed problems.Global polynomial and trigonometric bases are most effective in the smooth regime represented by the existing benchmarks.
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