Quantum AS-DeepONet for 2D PDEs

• Quantum AS-DeepONet is a hybrid quantum-classical neural operator that integrates stacked branch decomposition, PQCs, and cross-subnet attention for efficient 2D evolution equation solving.
• It embeds a hybrid quantum layer in the trunk and decentralized quantum subnets in the branch to preserve standard inner-product operators while reducing parameter count.
• Benchmark comparisons on 2D linear advection and Burgers’ equations show competitive error metrics using roughly 55–60% of the parameters of classical models.

Quantum AS-DeepONet is a hybrid quantum-classical neural operator for learning solution operators of 2D evolution equations. It combines a DeepONet backbone with three architectural modifications: a stacked decomposition of the branch input, Parameterized Quantum Circuits (PQCs) embedded in both branch and trunk subnetworks, and a cross-subnet attention mechanism adapted from Efficient Channel Attention (ECA). In the formulation reported in "Quantum AS-DeepOnet: Quantum Attentive Stacked DeepONet for Solving 2D Evolution Equations," the method is intended to preserve the standard DeepONet inner-product operator representation while reducing trainable parameter count relative to a classical DeepONet baseline, using only about 55–60% of the parameters of a deeper classical model on the reported benchmarks (Wang et al., 28 Feb 2026).

1. Definition and conceptual position

Quantum AS-DeepONet extends DeepONet for settings in which the input function is sampled at many sensor points and the target is a 2D evolution equation on the space-time domain

$\Omega = [0,1]\times[0,1]\times[0,1].$

The model is designed around the observation that the branch network of a standard DeepONet becomes expensive when sensor inputs are high-dimensional. The proposed response is not to replace the entire architecture with a fully quantum model, but to construct a hybrid operator network in which classical preprocessing and postprocessing surround quantum feature maps implemented by PQCs (Wang et al., 28 Feb 2026).

The acronym "AS" denotes "Attentive Stacked." "Stacked" refers to the decomposition of the branch input into multiple subvectors processed by separate hybrid quantum subnets. "Attentive" refers to a low-overhead cross-subnet attention module that reweights subnet outputs before they are concatenated into the final branch representation. The resulting architecture keeps the standard DeepONet pairing between branch and trunk features:

$$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$

A frequent source of ambiguity is the relation between Quantum AS-DeepONet and earlier quantum DeepONet variants. The 2024 "Quantum DeepONet" paper proposes a quantum-assisted DeepONet in which branch and trunk subnetworks are replaced by quantum orthogonal neural networks (QOrthoNNs) using unary encoding, orthogonal quantum layers, and tomography, but it does not introduce an AS-DeepONet architecture in the usual sense (Xiao et al., 2024). By contrast, Quantum AS-DeepONet is specifically a quantum attentive stacked DeepONet for 2D evolution equations.

2. Architectural organization

The architecture retains the standard DeepONet decomposition into a branch net and a trunk net, but both are modified.

The trunk net uses a single hybrid quantum layer:

$\mathbf{t}(y) = \mathbf{h}(y),$

where

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$

Here, $\psi \in \mathcal{S}_{m,q}^\sigma$ is a classical affine + activation map, $\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$ is the PQC, and $\phi \in \mathcal{S}_{q,p}^\sigma$ is a second classical affine + activation map. The trunk therefore maps the low-dimensional spatio-temporal coordinate into the latent basis used by the decoder (Wang et al., 28 Feb 2026).

The branch net is the main architectural novelty. Instead of feeding all sensor values into a single encoder, the sampled input

$u_s = (u(x_1), \ldots, u(x_i)) \in \mathbb{R}^d$

is partitioned into $r$ sub-vectors

$$u_s^{(1)}, \ldots, u_s^{(r)}, \qquad \sum_{j=1}^r d_j = d.$$

For simplicity, the paper assumes

$$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$0

Each sub-vector is processed by its own hybrid quantum subnet:

$$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$1

These outputs are stacked into

$$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$2

This decomposition serves two roles. First, it prevents the branch network from remaining monolithic when the input sensor dimension is large. Second, it allows each quantum subnet to operate on a manageable block of the input. This suggests that the design is motivated as much by qubit-limited deployment constraints as by representational considerations.

3. Hybrid quantum layer and cross-subnet attention

The hybrid quantum layer is defined by inserting a PQC between classical maps. Its measured components are

$$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$3

with

$$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$4

In this notation, $$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$5 is the encoding block, $$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$6 are parameterized circuit blocks, $$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$7 is the observable on the $$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$8-th qubit, $$G(u)(y) = \langle \mathbf{b}(u_{s}), \mathbf{t}(y) \rangle := \sum_{k=1}^{p} b_k(u_{s}) \cdot t_k(y) \in \mathbb{R}.$$9 is the encoded input, and $\mathbf{t}(y) = \mathbf{h}(y),$0 are trainable parameters (Wang et al., 28 Feb 2026).

The attention mechanism acts on subnet outputs rather than on channels or tokens. The paper defines global average pooling by

$\mathbf{t}(y) = \mathbf{h}(y),$1

where $\mathbf{t}(y) = \mathbf{h}(y),$2, producing $\mathbf{t}(y) = \mathbf{h}(y),$3. The 1D convolution kernel size is chosen adaptively as

$\mathbf{t}(y) = \mathbf{h}(y),$4

with hyperparameters $\mathbf{t}(y) = \mathbf{h}(y),$5, and nearest odd integer projection.

Local interactions are encoded through a Toeplitz banded matrix $\mathbf{t}(y) = \mathbf{h}(y),$6:

$\mathbf{t}(y) = \mathbf{h}(y),$7

Attention weights are then computed as

$\mathbf{t}(y) = \mathbf{h}(y),$8

and applied to the subnet outputs through

$\mathbf{t}(y) = \mathbf{h}(y),$9

The final branch representation is the concatenation

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$0

The paper characterizes this attention block as very cheap, with only $$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$1 learnable parameters. A plausible implication is that the attention module is intended to restore global coordination lost by partitioning the input, without incurring the cost of heavier attention mechanisms.

4. Operator-learning formulation and PDE setting

Quantum AS-DeepONet preserves the standard operator-learning view of DeepONet:

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$2

What changes is the generation of the latent features: branch features come from $$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$3 hybrid quantum subnets followed by cross-subnet attention, while trunk features come from one hybrid quantum layer (Wang et al., 28 Feb 2026).

The reported benchmarks are 2D evolution equations with periodic boundary conditions. The first is the 2D linear advection equation,

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$4

with

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$5

The second is the 2D nonlinear Burgers’ equation,

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$6

For both PDEs, the initial condition $$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$7 is generated as a 2D Gaussian random field using a Fourier-transform construction with Gaussian power spectral density

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$8

For Burgers’ equation, the reported correlation lengths are

$$\mathbf{t}(y) = \mathbf{h}(y) = \phi\bigl(\mathbf{f}(\psi(y))\bigr)\in \mathbb{R}^p.$$9

The paper does not discuss source terms because the presented equations are source-free.

The training objective is described as standard supervised regression optimized with Adam, and the reported metrics are relative $\psi \in \mathcal{S}_{m,q}^\sigma$0 error and final loss. The paper does not provide a displayed equation for the loss, but it states that training minimizes the prediction error between predicted and true PDE solutions.

5. Training configuration and parameter efficiency

The reported training setup is explicit. The optimizer is Adam, the initial learning rate is 0.002, the learning-rate schedule is three-stage, the number of epochs is 60,000, evaluation is performed every 500 epochs, and both the training and test sets contain 10,000 samples (Wang et al., 28 Feb 2026).

The quantum model settings differ by benchmark. For advection, the paper uses a 10-qubit circuit; for Burgers, a 12-qubit circuit. In both cases, the quantum depth is 2 and the pre-/post-processing layer widths are 50. The classical baselines are two DeepONet variants: Classical Model 1 with width 50 and depth 4, and Classical Model 2 with width 50 and depth 2.

The paper’s parameter-efficiency claim is summarized as "only 60% of the trainable parameters." The detailed counts are:

Benchmark	Quantum AS-DeepONet (circuit-block)	Classical Model 1
Advection	13,292	24,251
Burgers	14,342	24,251

These ratios are reported as approximately $\psi \in \mathcal{S}_{m,q}^\sigma$1 for advection and $\psi \in \mathcal{S}_{m,q}^\sigma$2 for Burgers. The "60%" statement is therefore an approximate summary rather than a single exact benchmark figure.

The paper also gives circuit-ansatz parameter formulas from Sim et al.:

• Circuit 2 (Nearest-neighbour): $\psi \in \mathcal{S}_{m,q}^\sigma$3
• Circuit 6 (All-to-all): $\psi \in \mathcal{S}_{m,q}^\sigma$4
• Circuit 19 (Circuit-block): $\psi \in \mathcal{S}_{m,q}^\sigma$5

The corresponding two-qubit gate counts are:

• Nearest-neighbour: $\psi \in \mathcal{S}_{m,q}^\sigma$6
• All-to-all: $\psi \in \mathcal{S}_{m,q}^\sigma$7
• Circuit-block: $\psi \in \mathcal{S}_{m,q}^\sigma$8

This supports the paper’s characterization of circuit-block as an efficiency-oriented compromise and all-to-all as more expressive but more costly.

6. Quantitative results and comparative behavior

The main quantitative comparison spans three quantum circuit choices—nearest-neighbour, all-to-all, and circuit-block—and two classical DeepONet baselines. For the advection equation, the reported results are:

Method	Relative $\psi \in \mathcal{S}_{m,q}^\sigma$9 error	Last loss
Quantum circuit-block	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$0	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$1
Quantum nearest-neighbour	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$2	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$3
Quantum all-to-all	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$4	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$5
Classical Model 1	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$6	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$7
Classical Model 2	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$8	$\mathbf{f}: \mathbb{R}^q \to \mathbb{R}^q$9

For the Burgers equation, the reported results are:

Method	Relative $\phi \in \mathcal{S}_{q,p}^\sigma$0 error	Last loss
Quantum circuit-block	$\phi \in \mathcal{S}_{q,p}^\sigma$1	$\phi \in \mathcal{S}_{q,p}^\sigma$2
Quantum nearest-neighbour	$\phi \in \mathcal{S}_{q,p}^\sigma$3	$\phi \in \mathcal{S}_{q,p}^\sigma$4
Quantum all-to-all	$\phi \in \mathcal{S}_{q,p}^\sigma$5	$\phi \in \mathcal{S}_{q,p}^\sigma$6
Classical Model 1	$\phi \in \mathcal{S}_{q,p}^\sigma$7	$\phi \in \mathcal{S}_{q,p}^\sigma$8
Classical Model 2	$\phi \in \mathcal{S}_{q,p}^\sigma$9	$u_s = (u(x_1), \ldots, u(x_i)) \in \mathbb{R}^d$0

Several conclusions are explicitly drawn in the paper. For advection, the quantum circuit-block model is essentially on par with Classical Model 1 in error and loss while using far fewer parameters. For Burgers, the quantum all-to-all ansatz obtains the best error among the quantum models and is close to Classical Model 1. The nearest-neighbour ansatz is consistently the weakest. Classical Model 2, despite having a parameter count comparable to the quantum models, performs substantially worse, which the paper uses to argue that parameter count alone does not explain performance (Wang et al., 28 Feb 2026).

The paper also reports visual comparisons between predicted and true solution fields for both PDEs and states that they show good agreement. Because the detailed image values are not given in the source material, the quantitative tables remain the main basis for comparison.

7. Relation to adjacent quantum DeepONet research, limitations, and interpretation

Quantum AS-DeepONet belongs to a broader line of quantum operator-learning models, but its design priorities differ from those of adjacent work. The 2024 "Quantum DeepONet" paper accelerates DeepONet evaluation by replacing branch and trunk subnetworks with QOrthoNNs based on unary encoding, reconfigurable beam splitter gates, and tomography, and it emphasizes a reduction in feedforward complexity from $u_s = (u(x_1), \ldots, u(x_i)) \in \mathbb{R}^d$1 to $u_s = (u(x_1), \ldots, u(x_i)) \in \mathbb{R}^d$2 (Xiao et al., 2024). The 2026 "Conformalized Quantum DeepONet Ensembles" paper extends that QOrthoNN-based line with ensemble uncertainty quantification, adaptive conformal prediction, and Superposed Parameterized Quantum Circuits (SPQCs) for ensemble compression (Matlia et al., 1 May 2026). Quantum AS-DeepONet is different in emphasis: it is centered on stacked branch decomposition, PQC-based hybrid layers, and low-cost cross-subnet attention for 2D evolution equations.

The paper explicitly notes several limitations. The experiments are simulation-only and use the PennyLane simulator rather than real quantum hardware. Noise robustness is not tested. Only two 2D evolution equations are benchmarked. The branch input is evenly split into subnets for simplicity. Available qubits remain limited. The paper also states that the quantum model is slower to train than classical DeepONet on simulator because of classical-to-quantum data conversion, simulator overhead, and the limitations of current quantum simulation and hardware (Wang et al., 28 Feb 2026).

Several assumptions are built into the formulation: periodic boundary conditions, equal-block partitioning of the sensor input, divisibility of the latent dimension $u_s = (u(x_1), \ldots, u(x_i)) \in \mathbb{R}^d$3 by the number of subnets $u_s = (u(x_1), \ldots, u(x_i)) \in \mathbb{R}^d$4, and representability of the target operator through the standard DeepONet inner-product form. The stated future directions are deployment on actual quantum hardware, testing noise robustness, and broader validation of applicability and scalability.

Taken together, these details position Quantum AS-DeepONet as a parameter-efficient hybrid operator-learning architecture rather than as a demonstrated hardware-speedup result. This suggests that its immediate significance lies in architectural design for quantum-enhanced neural operators under NISQ-era constraints, especially when high-dimensional branch inputs make a monolithic DeepONet branch undesirable.