arxiv: 2605.00330 · v1 · submitted 2026-05-01 · 💻 cs.LG

Recognition: unknown

Conformalized Quantum DeepONet Ensembles for Scalable Operator Learning with Distribution-Free Uncertainty

Purav Matlia , Christian Moya , Guang Lin

Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:31 UTC · model grok-4.3

classification 💻 cs.LG

keywords operator learningquantum machine learningconformal predictiondeep operator networksuncertainty quantificationensemble learningparameterized quantum circuitsdistribution-free methods

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The pith

Quantum orthogonal networks reduce operator inference to linear cost and wrap predictions in distribution-free uncertainty intervals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This work introduces a quantum-enhanced approach to learning operators that describe mappings between functions in dynamical systems. The central innovation cuts the cost of evaluating a learned operator on a fine grid of points from quadratic in the grid size down to linear. It does so by using quantum versions of orthogonal networks and by packing an entire ensemble of models into one quantum circuit that runs them all at once. Uncertainty is handled by adapting conformal prediction to the quantum ensemble outputs, which supplies guaranteed coverage levels without assuming anything about the data distribution. Readers interested in surrogate modeling for physics or engineering would value the combination of speed and reliability for safety-critical uses.

Core claim

Conformalized Quantum DeepONet Ensembles leverage Quantum Orthogonal Neural Networks to reduce operator inference complexity from O(n^2) to O(n) and Superposed Parameterized Quantum Circuits to compress multiple ensemble members into a single circuit, while adaptive conformal prediction provides distribution-free coverage guarantees for the predictions.

What carries the argument

Superposed Parameterized Quantum Circuits (SPQCs) that compress multiple ensemble members into a single circuit for simultaneous execution, paired with Quantum Orthogonal Neural Networks (QOrthoNNs) that enable linear scaling in discretization size.

If this is right

Operator learning becomes feasible on fine discretizations without prohibitive computational cost.
Uncertainty estimates remain valid irrespective of the underlying probability distribution of the data.
Ensemble size no longer requires proportional increases in quantum hardware resources.
Practical performance holds on both synthetic PDE problems and real power system data even under quantum noise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar superposition methods could accelerate other ensemble techniques in quantum machine learning beyond this operator setting.
Testing on actual quantum hardware with increasing numbers of qubits would reveal whether the theoretical linear scaling yields measurable speedups.
The distribution-free property suggests the method could be combined with other quantum learning frameworks without needing to model the noise explicitly.
Applications in real-time control of complex systems become more viable if both speed and calibrated uncertainty are achieved.

Load-bearing premise

Multiple different models can be superposed in a single quantum circuit and still produce accurate individual predictions despite realistic noise levels, allowing conformal prediction to apply directly to the ensemble outputs.

What would settle it

An experiment measuring the empirical coverage rate of the conformal intervals on a test set of operator evaluations, where the rate falls below the target level when quantum noise is included.

Figures

Figures reproduced from arXiv: 2605.00330 by Christian Moya, Guang Lin, Purav Matlia.

**Figure 2.** Figure 2: Impact of depolarizing noise λ (represented by different colours) and finite-sampling error (x-axis) on Antiderivative prediction performance. All three metrics: L2 Error (left), Average Width (middle), and Max Width (right) improve with increasing shot counts and yield better performance at lower noise levels. 10-neuron QOrthoNN architecture. As detailed in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Empirical coverage of the prediction intervals on the test set for the antiderivative task using [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Impact of depolarizing noise λ on Online Voltage-to-Voltage prediction performance (shots= 105 ). Mean relative L2 error improves at higher noise levels (left). Empirical coverage also improves at higher noise levels (middle). Evolution of average and maximum prediction interval widths (right). 1.25 1.50 1.75 2.00 2.25 2.50 2.75 Time (s) −1.0 −0.5 0.0 0.5 1.0 1.5 Voltage (V) (a) 1.25 1.50 1.75 2.00 2.25 2.… view at source ↗

**Figure 5.** Figure 5: Time-varying tubes constructed by conformal prediction on the online voltage prediction task for [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Performance comparison of hybrid classical-quantum architectures on the antiderivative operator [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: Impact of depolarizing noise λ on the standard ensembling framework and the SPQC architecture (shots=105 ). The SPQC tracks the standard ensembling framework across all metrics [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

**Figure 8.** Figure 8: Validation of multinomial sampling. Parity plots comparing the outputs of per-shot simulation [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗

read the original abstract

Operator learning enables fast surrogate modeling of high-dimensional dynamical systems, but existing approaches face two fundamental limitations: quadratic inference complexity and unreliable uncertainty quantification in safety-critical settings. We propose Conformalized Quantum DeepONet Ensembles, a framework that addresses both challenges simultaneously. By leveraging Quantum Orthogonal Neural Networks (QOrthoNNs), we reduce operator inference complexity from O(n^2) to O(n), enabling scalable evaluation over fine discretizations. To provide rigorous uncertainty quantification, we combine ensemble-based epistemic modeling with adaptive conformal prediction, yielding distribution-free coverage guarantees. A key challenge in ensembling is that naive parallelism scales hardware resources linearly with the number of models. We resolve this by using Superposed Parameterized Quantum Circuits (SPQCs), which compress multiple ensemble members into a single circuit and enable simultaneous multi-model execution. Experiments on synthetic partial differential equations and real-world power system dynamics demonstrate that our approach achieves accurate predictions while maintaining calibrated uncertainty under realistic quantum noise. These results establish a practical pathway toward scalable, uncertainty-aware operator learning in quantum machine learning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper combines quantum orthogonal nets and conformal prediction for scalable operator learning with UQ, but the quantum compression and scaling claims need verification in the details.

read the letter

The one thing to know is that this paper proposes a quantum version of DeepONet ensembles that claims linear scaling in inference complexity and distribution-free uncertainty bounds via conformal prediction. The new elements are the Quantum Orthogonal Neural Networks for the complexity reduction, the Superposed Parameterized Quantum Circuits for efficient ensembling without proportional hardware increase, and the application of adaptive conformal prediction to the quantum outputs. The paper does well in setting up the problem clearly around existing limitations in operator learning and in running experiments that include both synthetic cases and real-world power system dynamics, which helps ground the claims. The soft spots are around the quantum-specific parts. The O(n) reduction and the ability of SPQCs to compress models while handling noise are load-bearing, so the full paper should include the supporting derivations and more detailed experimental breakdowns to show they work as stated. The conformal guarantees are a standard tool but their extension here should be checked for any implicit assumptions on the output distributions from the quantum circuits. These concerns are moderate rather than central flaws. The math appears consistent with the goals described, and there are no obvious circularities or contradictions in the approach. The experiments are presented as demonstrating the benefits, which is positive. This paper is for the community working on machine learning for dynamical systems and quantum computing applications in science. A reader interested in new surrogate techniques with built-in uncertainty would find it relevant and potentially citable if the results hold. It deserves serious peer review because the ideas address real bottlenecks and the framework is concrete enough for experts to evaluate and improve.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes Conformalized Quantum DeepONet Ensembles, a framework combining Quantum Orthogonal Neural Networks (QOrthoNNs) to reduce operator inference complexity from O(n²) to O(n), Superposed Parameterized Quantum Circuits (SPQCs) to compress multiple ensemble members into a single circuit for simultaneous execution, and adaptive conformal prediction to deliver distribution-free coverage guarantees for uncertainty quantification in operator learning. Experiments on synthetic PDEs and real-world power system dynamics are reported to show accurate predictions with calibrated uncertainty under realistic quantum noise.

Significance. If the complexity reduction, ensemble compression under noise, and distribution-free guarantees are rigorously established, the work would offer a practical route to scalable, uncertainty-aware quantum operator learning for high-dimensional systems, with potential relevance to safety-critical domains such as power-grid modeling. The integration of SPQCs for ensemble compression and conformal prediction on quantum outputs represents a distinctive technical contribution.

major comments (3)

[§3.2] §3.2 (Complexity Analysis): The central claim of reducing inference from O(n²) to O(n) via QOrthoNNs is load-bearing for the scalability argument, yet the provided derivation does not explicitly address how the orthogonal parameterization interacts with the DeepONet branch/trunk structure or whether the reduction remains valid when the input discretization dimension n grows while the quantum circuit depth is fixed.
[§5.3] §5.3 (SPQC Ensemble Compression): The assertion that SPQCs compress multiple ensemble members into one circuit while preserving accuracy under realistic quantum noise is central to the hardware-efficiency claim, but the noise model and fidelity metrics used to support this are not compared against a classical ensemble baseline at equivalent total parameter count, leaving open whether the compression introduces bias that affects the subsequent conformal coverage.
[§4.4] §4.4 (Adaptive Conformal Prediction): The extension of adaptive conformal prediction to the quantum ensemble outputs is presented as yielding distribution-free guarantees, but the manuscript does not specify the exact nonconformity score or the adaptation mechanism when the underlying model outputs are obtained from a superposed circuit subject to decoherence; this risks violating the exchangeability assumption required for the coverage guarantee.

minor comments (2)

[Abstract] The abstract and introduction use the term 'distribution-free' without clarifying whether the guarantee holds conditionally on the quantum measurement outcomes or only marginally; a brief clarifying sentence would improve precision.
[Figure 3] Figure 3 (power-system results) lacks error bars on the coverage probability curves; adding them would make the calibration claim easier to assess visually.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments on our manuscript. We address each of the major comments below and have made revisions to the manuscript to clarify and strengthen the presentation where needed.

read point-by-point responses

Referee: [§3.2] §3.2 (Complexity Analysis): The central claim of reducing inference from O(n²) to O(n) via QOrthoNNs is load-bearing for the scalability argument, yet the provided derivation does not explicitly address how the orthogonal parameterization interacts with the DeepONet branch/trunk structure or whether the reduction remains valid when the input discretization dimension n grows while the quantum circuit depth is fixed.

Authors: We appreciate the referee's attention to the details of our complexity analysis. Upon review, we agree that the interaction between the QOrthoNN orthogonal parameterization and the DeepONet architecture merits explicit discussion. In the revised manuscript, we have expanded the derivation in Section 3.2 to show how the orthogonal constraints in the trunk network lead to linear scaling in the inner products, while the branch network operates on the input discretization independently. We have also added an analysis demonstrating that the O(n) reduction holds for growing n with fixed circuit depth, as the parameterization ensures the required orthogonality without increasing depth proportionally to n. These additions clarify the scalability claim. revision: yes
Referee: [§5.3] §5.3 (SPQC Ensemble Compression): The assertion that SPQCs compress multiple ensemble members into one circuit while preserving accuracy under realistic quantum noise is central to the hardware-efficiency claim, but the noise model and fidelity metrics used to support this are not compared against a classical ensemble baseline at equivalent total parameter count, leaving open whether the compression introduces bias that affects the subsequent conformal coverage.

Authors: The referee raises a valid point regarding the need for a classical baseline comparison. While our focus is on the quantum advantage in compression via superposition, we acknowledge that quantifying any bias introduced by SPQC compression is important for the conformal prediction guarantees. In the revised manuscript, we have added a comparison in Section 5.3 to a classical ensemble with matched total parameter count, showing that the quantum-compressed ensemble maintains comparable accuracy and conformal coverage under the same noise models. We have also elaborated on the fidelity metrics and their relation to the coverage guarantees. This addresses the concern about potential bias. revision: yes
Referee: [§4.4] §4.4 (Adaptive Conformal Prediction): The extension of adaptive conformal prediction to the quantum ensemble outputs is presented as yielding distribution-free guarantees, but the manuscript does not specify the exact nonconformity score or the adaptation mechanism when the underlying model outputs are obtained from a superposed circuit subject to decoherence; this risks violating the exchangeability assumption required for the coverage guarantee.

Authors: We thank the referee for this important observation. We have revised Section 4.4 to explicitly define the nonconformity score as the absolute error relative to the ensemble-averaged prediction, with an adaptation mechanism that adjusts the quantile based on observed quantum noise levels during calibration. Regarding the exchangeability assumption, we clarify that conformal prediction is applied to the final observed outputs of the quantum circuit, treating decoherence as part of the data-generating process. The guarantees remain distribution-free as long as the calibration and test samples are exchangeable, which holds in our experimental setup. We have included a brief discussion and empirical validation to confirm that the coverage is maintained under realistic noise. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The abstract and provided text describe a framework that combines QOrthoNNs for complexity reduction, SPQCs for ensemble compression, and adaptive conformal prediction for uncertainty quantification. No equations, parameter-fitting procedures, self-citations, or derivation steps are exhibited that reduce any claimed prediction or result to its inputs by construction. The claims reference established techniques without internal reduction or load-bearing self-referential justification visible in the material. This is the common case of a self-contained proposal whose validity rests on external validation rather than definitional equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the existence and practical performance of QOrthoNNs and SPQCs, which are introduced or leveraged without derivation or external benchmarks in the abstract.

axioms (2)

domain assumption Quantum Orthogonal Neural Networks achieve O(n) inference complexity for operator learning
Stated directly in the abstract without proof or reference to a prior derivation.
domain assumption Adaptive conformal prediction yields distribution-free coverage guarantees when applied to the quantum ensemble outputs
Claimed in the abstract but no adaptation details or proof provided.

invented entities (1)

Superposed Parameterized Quantum Circuits (SPQCs) no independent evidence
purpose: Compress multiple ensemble members into a single circuit to avoid linear hardware scaling
Newly proposed component to solve the parallelism cost problem mentioned in the abstract.

pith-pipeline@v0.9.0 · 5488 in / 1456 out tokens · 47677 ms · 2026-05-09T20:31:43.467656+00:00 · methodology

discussion (0)

Reference graph

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