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

Muhammad Abid; Arth Sojitra; Bipin Tiwari; Omer San

quant-phAug 27, 2026

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

Authors: Muhammad Abid, Arth Sojitra, Bipin Tiwari, Omer San

Organizations: Department of Mechanical and Aerospace Engineering University of Tennessee, Knoxville Knoxville, TN 37996, USA

Abstract

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.

Figures & tables

Figure 1: The circuit of a single quantum layer, comprising data loading, a pyramidal orthogonal transformation, and tomography, with one ancillary qubit for the tomography. Vertical connectors denote the two-qubit RBS gates; $\theta_{i}$ parameterize the pyramid and $\alpha_{i}$ the loader. The top panels show the pyramid geometry for representative weight shapes $\mathbf{W}\in\mathbb{R}^{m\times n}$ .

Figure 2: The Quantum SEDONet architecture. As in Quantum DeepONet, a branch net encodes the input function $v$ and a trunk net encodes the query coordinate $\xi$ ; each subnetwork is a stack of QOrthoNN quantum layers (Load $\to$ Pyramid $\to$ Tomography, with classical $\sigma$ between layers and a classical output layer), and the two outputs are combined by an inner product to give $\mathcal{G}^{\prime}(v)(\xi)$ . Quantum SEDONet adds a single new block, the highlighted spectral embedding, which maps the trunk coordinate $\xi$ to $\phi(\xi)$ using a Fourier basis for periodic coordinates and a Chebyshev basis for bounded non-periodic ones before the quantum layers.

Panel of Figure 3 — Figure 3: Ideal quantum simulation of function approximation with a QOrthoNN. Black, red, and blue denote the reference function, the classical OrthoNN prediction, and the ideal quantum simulation of the QOrthoNN respectively; the three curves are superimposed by construction. ( a ) $f(x)=1/(1+25x^{2})$ on $[-1,1]$ , relative $L^{2}$ error $0.136\%$ , classical–quantum agreement $7.8\times 10^{-9}$ . ( b ) $f(x)=\sum_{k=1}^{4}\sin(kx)$ on $[-\pi,\pi]$ , relative $L^{2}$ error $3.308\%$ , agreement $1.0\times 10^{-6}$ .

Panel of Figure 4 — Figure 4: Antiderivative operator, both arms at $11$ trunk qubits. ( a ) Per-function comparison over the $100$ test functions, each point giving the relative $L^{2}$ error of the two arms on the same input function; points below the diagonal favor Quantum SEDONet, which is more accurate on $99$ of $100$ functions. ( b ) The test function on which the baseline performs worst. The Quantum DeepONet prediction is piecewise linear with visible hinge points, the signature of a ReLU trunk driven by a single affine coordinate; the Chebyshev embedding removes them. This sample has roughly $20\times$ smaller amplitude than a typical test function, which inflates its relative error.

Figure 5: Prediction comparison for one representative advection test sample, showing the initial condition, the ground truth, Quantum DeepONet and Quantum SEDONet predictions, and their corresponding absolute error maps on a common color scale. The baseline error follows the diagonal transport characteristics; the spectral embedding removes it.

Figure 6: Advection: mean relative $L^{2}$ error over $200$ test functions, with standard-deviation bars. Quantum SEDONet improves on the baseline for $199$ of $200$ functions (paired $t=23.3$ ).

Figure 7: Prediction comparison for one representative Burgers equation test sample, showing the initial condition, the ground truth, Quantum DeepONet and Quantum SEDONet predictions, and their corresponding absolute error maps on a common color scale.

Figure 8: Burgers: mean relative $L^{2}$ error over $100$ test functions, with standard-deviation bars. Quantum SEDONet improves on the baseline for $93$ of $100$ functions (paired $t=8.1$ ).

Figure 9: Prediction comparison for one representative channel Poisson test sample, showing the source field, the ground truth, Quantum DeepONet and Quantum SEDONet predictions, and their corresponding absolute error maps on a common color scale. The solution is periodic in $x$ and satisfies homogeneous Dirichlet conditions at $y=0,1$ . Quantum SEDONet reduces the error where the wall-normal structure is sharpest.

Figure 10: Channel Poisson: mean relative $L^{2}$ error over the test functions, with standard-deviation bars. Quantum SEDONet improves on the baseline for $98\%$ of functions (paired $t=26.8$ ).

Benchmark	Q-DeepONet	Q-SEDONet	Reduction	Better on
Antiderivative	$1.159\%$	$0.532\%$	$54.1\%$	$99/100$
Advection	$3.463\%$	$1.744\%$	$49.6\%$	$199/200$
Burgers	$2.313\%$	$1.481\%$	$36.0\%$	$93/100$
Channel Poisson	$5.889\%$	$3.760\%$	$36.2\%$	$98/100$

Table 1: Mean relative

L^{2}

error, baseline versus Quantum SEDONet. “Better on” counts test functions on which Quantum SEDONet has the lower error under a paired comparison on identical inputs; the corresponding paired

t

-statistics are reported in Sections 4.4 , 4.5 and 4.6 . Results for the antiderivative operator, whose embedding is Chebyshev alone, are given in Section 4.3 .

Benchmark	Baseline trunk	SEDONet trunk	Qubits	Parameters
Advection	raw $(x,t)$	Fourier $(x)+$ Cheb $(t)$	$22\to 22$	$3544\to 3669$ ( $+3.5\%$ )
Burgers	raw $(x,t)$	Fourier $(x)+$ Cheb $(t)$	$21\to 21$	$2870\to 2940$ ( $+2.4\%$ )
Channel Poisson	raw $(x,y)$	Fourier $(x)+$ Cheb $(y)$	$21\to 21$	$2769\to 2884$ ( $+4.2\%$ )

Table 2: Resource cost of the spectral trunk embedding. Qubit count and circuit depth are unchanged on every benchmark. The antiderivative operator, whose trunk register is likewise unchanged at

11

qubits, is described in Section 4.3 .

PDE Operator Learning

Partial differential equation (PDE) operator learning builds models that map inputs such as coefficients and initial conditions to solution fields.

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Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Muhammad Abid, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Deep Operator Networks

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Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative

L^2

errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.

Abstract

PDE Operator Learning

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