Conformalized Quantum DeepONet Ensembles

• The paper proposes a quantum operator-learning framework that reduces quadratic layer complexity to linear, enabling scalable evaluations.
• It replaces classical orthogonal layers with QOrthoNNs and compresses ensemble models using SPQCs, significantly lowering qubit requirements.
• Adaptive conformal prediction calibrates ensemble outputs to offer distribution-free uncertainty guarantees with provable finite-sample coverage.

Conformalized Quantum DeepONet Ensembles are a quantum machine learning framework for operator learning that combines Quantum Orthogonal Neural Networks (QOrthoNNs), ensemble-based epistemic modeling, adaptive conformal prediction, and Superposed Parameterized Quantum Circuits (SPQCs) in order to address two limitations identified for existing approaches: quadratic inference complexity and unreliable uncertainty quantification in safety-critical settings (Matlia et al., 1 May 2026). In the formulation reported in the literature, the framework targets scalable evaluation over fine discretizations by reducing operator inference complexity from $\mathcal O(n^2)$ to $\mathcal O(n)$, while also providing distribution-free coverage guarantees through conformal prediction. The reported empirical setting includes synthetic partial differential equations and real-world power system dynamics, with uncertainty behavior examined under realistic quantum noise (Matlia et al., 1 May 2026).

1. Operator-learning setting and the DeepONet baseline

The underlying task is operator learning, in which an operator

$$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$

is approximated by a DeepONet architecture. In the stated construction, DeepONet separates the computation into a branch network and a trunk network,

$$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$

followed by a dot-product readout,

$$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$

The scaling bottleneck arises from the dense layers used in the branch or trunk components. For a network of hidden width $n$, each dense layer costs $\mathcal O(n^2)$. When the trunk must be evaluated $M$ times for $M$ query points, the per-input cost becomes $\mathcal O(M\,n^2)$. The source text explicitly identifies this as prohibitive for large $\mathcal O(n)$0 and for extensive Monte-Carlo runs in the range $\mathcal O(n)$1–$\mathcal O(n)$2 scenarios (Matlia et al., 1 May 2026).

The motivation for a quantum implementation is stated in three parts. First, parameterized quantum circuits can in principle emulate orthogonal transforms in sub-quadratic depth. Second, quantum uncertainty in the form of shot noise does not by itself capture epistemic model uncertainty. Third, naive quantum ensembling multiplies qubit requirements by the ensemble size $\mathcal O(n)$3. This combination motivates a joint quantum-plus-conformal design rather than a purely quantum surrogate or a purely classical uncertainty wrapper.

2. QOrthoNNs and sub-quadratic operator inference

QOrthoNNs are introduced as the mechanism for replacing a classical orthogonal layer

$\mathcal O(n)$4

with a quantum circuit implementation (Matlia et al., 1 May 2026). The reported construction uses $\mathcal O(n)$5 data qubits $\mathcal O(n)$6 plus one ancilla and consists of three stages:

1. $\mathcal O(n)$7, an amplitude-loader into the unary subspace.
2. $\mathcal O(n)$8, a pyramidal RBS (Reconfigurable Beam Splitter) network implementing $\mathcal O(n)$9.
3. Tomography plus ancilla manipulation to read out each $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$0.

The complexity claim is central. Classically, each orthogonal layer remains $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$1. In the quantum realization, the data loader depth is stated as $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$2, the pyramidal unitary depth as $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$3, and tomography as $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$4 using two loaders plus $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$5 gates. The resulting total depth is therefore $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$6. Because amplitude encoding and tomography each require $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$7 shots for error $$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$8, the end-to-end cost is

$$\mathcal G: u(\cdot)\mapsto y = \mathcal G[u](\cdot)$$9

which is described as sub-quadratic in $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$0 (Matlia et al., 1 May 2026).

This reduction is significant specifically for operator-learning workloads in which fine discretizations force repeated trunk evaluations. A plausible implication is that the advantage is most relevant when the cost of repeated classical dense-layer application dominates the overall surrogate evaluation budget.

3. SPQCs and ensemble compression

The framework does not rely on a single QOrthoNN model. It uses ensembles to represent epistemic uncertainty, but it also addresses the hardware cost of quantum ensembling. The baseline comparison is explicit: $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$1 independent QOrthoNN models in parallel require $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$2 qubits, whereas an SPQC encodes all $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$3 models in one circuit using $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$4 address qubits (Matlia et al., 1 May 2026).

The SPQC state preparation is given by

$$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$5

In this construction, the address register $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$6 selects model $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$7, and controlled unitaries conditionally load data $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$8 and apply $$\mathbf b = \mathrm{Branch}_\theta(u)\in\mathbb R^p, \qquad \mathbf t = \mathrm{Trunk}_\theta(y)\in\mathbb R^p,$$9.

The extraction mechanism is also specified: after a single joint measurement, with post-selection to unary outcomes, all $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$0 model outputs are obtained simultaneously. The circuit depth scales as $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$1 because of controlled-RBS ladders, while the qubit count is only

$$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$2

The framework therefore trades linear qubit growth for logarithmic address overhead at the cost of circuit-depth growth in $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$3. This trade-off is later reflected in the stated limitations: SPQC depth $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$4 can accumulate noise for large $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$5. A common misconception addressed directly by this design is that quantum parallelism alone makes ensembling effectively free; in the reported formulation, naive parallelism still scales hardware resources linearly with the number of models.

4. Conformal prediction and distribution-free uncertainty

The uncertainty layer combines ensemble statistics with adaptive conformal prediction (Matlia et al., 1 May 2026). For each calibration point $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$6, the ensemble mean $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$7 and standard deviation $$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$8 are computed, and the adaptive nonconformity score is defined as

$$\mathcal G_\theta[u](y)\;=\;\mathbf b^\top\mathbf t.$$9

Let $n$0 denote the $n$1-quantile of the calibration scores $n$2. For a new input $n$3, the prediction set is

$n$4

Under the standard conformal assumption of exchangeability between calibration and test data, the framework has the finite-sample guarantee

$n$5

The conceptual role of conformalization is explicit. Quantum uncertainty arising from shot noise alone is not treated as a sufficient account of model uncertainty, so the framework retains model ensembles and then calibrates their outputs with conformal prediction. This separates epistemic modeling from finite-sample calibration. The reported guarantee is distribution-free, but only under exchangeability; the same source notes that this assumption is violated in streaming and temporal tasks, which is an important qualification rather than a minor technical caveat.

5. Reported empirical results

The reported experiments cover synthetic PDE benchmarks and real-world power system benchmarks, with a target coverage of $n$6 throughout (Matlia et al., 1 May 2026).

For the synthetic PDE setting, two benchmark operators are listed: the antiderivative $n$7 with $n$8, $n$9, and the one-dimensional advection equation $\mathcal O(n^2)$0 with periodic boundary conditions.

The ideal simulation results are:

• Antiderivative, $\mathcal O(n^2)$1: Relative $\mathcal O(n^2)$2 error $\mathcal O(n^2)$3, coverage $\mathcal O(n^2)$4, average width $\mathcal O(n^2)$5, peak uncertainty $\mathcal O(n^2)$6.
• Antiderivative, $\mathcal O(n^2)$7: Relative $\mathcal O(n^2)$8 error $\mathcal O(n^2)$9, coverage $M$0, average width $M$1, peak uncertainty $M$2.
• Advection, $M$3: Relative $M$4 error $M$5, coverage $M$6, average width $M$7, peak uncertainty $M$8.
• Advection, $M$9: Relative $M$0 error $M$1, coverage $M$2, average width $M$3, peak uncertainty $M$4.

For the power system setting, three tasks are listed: offline voltage-to-voltage forecasting $M$5 from $M$6 pre-fault to $M$7 post-fault, offline voltage-to-active-power $M$8, and online sliding-window voltage-to-voltage $M$9.

The ideal simulation results are:

• Online V→V: Relative $\mathcal O(M\,n^2)$0 error $\mathcal O(M\,n^2)$1, coverage $\mathcal O(M\,n^2)$2, average width $\mathcal O(M\,n^2)$3, peak uncertainty $\mathcal O(M\,n^2)$4.
• Offline V→V: Relative $\mathcal O(M\,n^2)$5 error $\mathcal O(M\,n^2)$6, coverage $\mathcal O(M\,n^2)$7, average width $\mathcal O(M\,n^2)$8, peak uncertainty $\mathcal O(M\,n^2)$9.
• Offline V→P: Relative $\mathcal O(n)$00 error $\mathcal O(n)$01, coverage $\mathcal O(n)$02, average width $\mathcal O(n)$03, peak uncertainty $\mathcal O(n)$04.

Taken together, these experiments are presented as demonstrating accurate predictions with calibrated uncertainty. This suggests that the conformal layer remains effective across both synthetic operator benchmarks and application-driven dynamical-system surrogates, although the reported accuracy and interval widths vary substantially by task.

6. Quantum noise, limitations, and stated future directions

The impact of quantum noise is explicitly examined under hardware-calibrated depolarizing and readout noise from IBM Marrakesh, Torino, and Brisbane (Matlia et al., 1 May 2026). The reported finding is that empirical coverage remains at least $\mathcal O(n)$05 across shot budgets from $\mathcal O(n)$06 to $\mathcal O(n)$07. The same source additionally reports that, for very shallow circuits with at most $\mathcal O(n)$08 qubits, increased depolarizing noise can act as a regularizer, slightly improving coverage and sharpening intervals.

The strengths claimed for the framework are equally explicit: sub-quadratic inference $\mathcal O(n)$09 via QOrthoNN, provable distribution-free uncertainty via conformal prediction, and logarithmic-scale qubit cost in ensembles via SPQC. The limitations are also directly enumerated. Conformal validity assumes calibration/test exchangeability and is violated in streaming or temporal tasks; near-term quantum devices remain noisy, so classical-quantum hybrid variants may be needed; and SPQC depth $\mathcal O(n)$10 can accumulate noise for large ensemble sizes.

The future directions listed in the source are: adaptive conformal methods for non-exchangeable data, including weighted and Mondrian variants; theoretical study of noise-induced regularization in shallow PQCs; hardware demonstrations on more than $\mathcal O(n)$11 qubits as coherence improves; and multi-fidelity hybrid architectures combining a classical branch, a quantum trunk, and an SPQC ensemble. A plausible implication is that subsequent work will focus less on the existence of a conformalized quantum operator-learning pipeline and more on robustness under temporal shift, hardware noise accumulation, and hybrid architectural decomposition.