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arxiv: 2605.15410 · v1 · pith:I5TLESTCnew · submitted 2026-05-14 · 🪐 quant-ph · cs.AI· cs.LG

Diagonal Adaptive Non-local Observables on Quantum Neural Networks

Huan-Hsin Tseng , Yan Li , Hsin-Yi Lin , Samuel Yen-Chi Chen This is my paper

Pith reviewed 2026-05-19 15:16 UTC · model grok-4.3

classification 🪐 quant-ph cs.AIcs.LG
keywords adaptive non-local observablesdiagonal observablesquantum neural networksvariational quantum algorithmsmeasurement designfunction space enlargement
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The pith

Diagonal observables in adaptive non-local setups match full ANO power while cutting complexity from O(4^k) to O(2^k).

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

The paper proposes restricting adaptive non-local observables to diagonal form when paired with quantum circuits. This restriction is presented as mathematically equivalent to the full space of general Hermitian observables, because diagonal matrices serve as canonical representatives modulo unitary similarity. A sympathetic reader would care because the change keeps the enlargement of the function space available to variational quantum algorithms yet lowers the parameter count and classical optimization cost on the measurement side. The resulting diagonal ANO includes ordinary variational quantum circuits as the special case of a fixed observable.

Core claim

Diagonal ANO retains the same capability of full ANO while reducing k-local observable complexity from O(4^k) to O(2^k) and lowering the corresponding measurement-side classical computation. Mathematically, this is equivalent to the full ANO of a large parameter space since diagonal matrices are canonical representatives of the ANO space modulo unitary similarity. In this sense, diagonal ANO preserves much of the benefit of full ANO while encompassing conventional VQCs as a special case.

What carries the argument

Diagonal observables paired with quantum circuits, acting as canonical representatives of the ANO space modulo unitary similarity.

Load-bearing premise

Restricting to diagonal observables paired with quantum circuits is mathematically equivalent to the full ANO space without loss of function-space enlargement in the variational setting.

What would settle it

A concrete calculation or numerical test showing that some variational task achievable with general ANO cannot be matched by any diagonal ANO with the same circuit family.

Figures

Figures reproduced from arXiv: 2605.15410 by Hsin-Yi Lin, Huan-Hsin Tseng, Samuel Yen-Chi Chen, Yan Li.

Figure 2
Figure 2. Figure 2: The training process searching for optimal circuit [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: A VQC diagram of (11), (12), which is also imple [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Observable space H(n) is partitioned into equivalent classes labeled by R 2 n . In other words, H(n) is classified by its eigenvalues, and VQC falls in only one particular class. By this observation, we propose using a diagonalization pair (Λ, U) to represent an ANO of (4) and (5), thereby reducing the number of Hermitian parameters from K2 to K, which is called Diagonal Adaptive Non-local Observables (DAN… view at source ↗
Figure 5
Figure 5. Figure 5: shows the test accuracies over epochs, and Table I summarizes the best test accuracies and parameter counts. Since the VQC model is fixed across all settings, the observed improvements are attributable to the added variety from the dynamical diagonal observables Λ(λ). The empty entries of ANO 6-local and 8-local in Table I are due to simulations exceeding the available 128GB RAM in our statevector implemen… view at source ↗
Figure 6
Figure 6. Figure 6: Yale B: 10 individuals selected for classification. Each cropped image (192 × 168 pixels) is flattened, stan￾dardized, and reduced to 16 dimensions via PCA. The resulting features are linearly rescaled to [−π, π] for angle encoding, and a stratified split of 1,584/198/198 samples is used for training/validation/test, respectively. The choice of PCA serves to reduce data dimension while introducing no addit… view at source ↗
Figure 8
Figure 8. Figure 8: Yale B test accuracy curves over epochs. Performance steadily improved by increasing non-locality k. TABLE II: k-local DANO on Extended Yale B from [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reconstruction of first 5 individuals in [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Additional results. The branch with dotted line is by switching from pure VQC to DANO of k = 10 only the measurement eigenvalues to change. Eventually reaching 80.81%, approaching the original 10-local 87.9%. V. CONCLUSION We introduced DANO, in which a variational circuit U(θ) is paired with trainable diagonal observables Λ(λ). Mathematically, diagonal observables serve as canonical representatives of ANO… view at source ↗
read the original abstract

Adaptive Non-local Observables (ANOs) have shown that making quantum observables dynamic can substantially enlarge the function space of Variational Quantum Algorithms, partly shifting hardware demands from circuit synthesis to measurement design. However, this advantage is accompanied by a steep increase in the number of parameters, as well as the classical optimization cost for varying general Hermitian observables. We propose a special form of ANO that significantly reduces this burden by considering only diagonal observables paired with quantum circuits. Mathematically, this is equivalent to the full ANO of a large parameter space since diagonal matrices are canonical representatives of the ANO space modulo unitary similarity. As a result, Diagonal ANO retains the same capability of full ANO while reducing $k$-local observable complexity from $O(4^k)$ to $O(2^k)$ and lowering the corresponding measurement-side classical computation. In this sense, diagonal ANO preserves much of the benefit of full ANO while encompassing conventional VQCs as a special case.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Diagonal Adaptive Non-local Observables (Diagonal ANO) for quantum neural networks. It asserts that restricting observables to diagonal form is mathematically equivalent to the full space of Adaptive Non-local Observables (ANOs) because diagonal matrices serve as canonical representatives modulo unitary similarity. This restriction is claimed to retain the full capability of ANOs while reducing k-local observable complexity from O(4^k) to O(2^k), lowering measurement-side classical computation, and encompassing conventional variational quantum circuits (VQCs) as a special case.

Significance. If the claimed equivalence holds for typical fixed-depth circuit ansatze, the result would meaningfully advance practical use of adaptive observables by cutting parameter count and classical overhead without sacrificing the function-space enlargement demonstrated for general ANOs. The reduction in measurement complexity and the inclusion of standard VQCs as a limit case would be concrete strengths, provided the variational function space is preserved.

major comments (2)
  1. [Abstract] Abstract: the assertion that 'diagonal matrices are canonical representatives of the ANO space modulo unitary similarity' and therefore yield mathematical equivalence is not accompanied by an explicit mapping, derivation, or verification that the reachable function space remains unchanged when the circuit ansatz is fixed (e.g., hardware-efficient layers of limited depth). The unitary V that diagonalizes a general Hermitian O must be absorbed into the circuit; for a restricted parameterization this absorption is not guaranteed, so the set of representable functions with (θ, diagonal entries) is generally a strict subset of those with (θ, general Hermitian).
  2. [Abstract] Abstract: the claim that Diagonal ANO 'retains the same capability of full ANO' therefore rests on an unverified assumption about ansatz universality. No section demonstrates that the enlarged function space of ANOs survives the restriction for the concrete circuit families used in the numerical experiments or analysis.
minor comments (1)
  1. [Abstract] Notation for the complexity reduction O(4^k) to O(2^k) should be accompanied by a brief definition of what 'k-local observable complexity' counts (number of independent real parameters, number of measurement bases, or something else).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and insightful comments on our manuscript. We address each major comment point by point below, providing the strongest honest clarification of our claims while outlining revisions where the presentation can be strengthened.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that 'diagonal matrices are canonical representatives of the ANO space modulo unitary similarity' and therefore yield mathematical equivalence is not accompanied by an explicit mapping, derivation, or verification that the reachable function space remains unchanged when the circuit ansatz is fixed (e.g., hardware-efficient layers of limited depth). The unitary V that diagonalizes a general Hermitian O must be absorbed into the circuit; for a restricted parameterization this absorption is not guaranteed, so the set of representable functions with (θ, diagonal entries) is generally a strict subset of those with (θ, general Hermitian).

    Authors: We agree that an explicit derivation would improve clarity. By the spectral theorem, any Hermitian O admits a decomposition O = V D V† with D diagonal. The expectation value then satisfies ⟨ψ(θ)|O|ψ(θ)⟩ = ⟨V†ψ(θ)|D|V†ψ(θ)⟩, showing that a diagonal observable on the rotated state is equivalent. Because the circuit parameters θ are variationally optimized, the ansatz can absorb the action of V whenever it is sufficiently expressive. We acknowledge, however, that for strictly depth-limited hardware-efficient ansatze the absorption need not be exact and the representable set may be a proper subset. In the revision we will insert a dedicated paragraph deriving the mapping from the spectral theorem and add a brief discussion of the expressivity conditions required for equivalence. revision: partial

  2. Referee: [Abstract] Abstract: the claim that Diagonal ANO 'retains the same capability of full ANO' therefore rests on an unverified assumption about ansatz universality. No section demonstrates that the enlarged function space of ANOs survives the restriction for the concrete circuit families used in the numerical experiments or analysis.

    Authors: The manuscript asserts equivalence on the basis of the canonical-representative property modulo unitary similarity, yet we did not supply a separate verification that the function-space enlargement observed for general ANOs is preserved under the specific fixed-depth families appearing in our numerics. We will therefore expand the methods section with a short theoretical argument linking the spectral mapping to the variational optimization and, where feasible, include a brief supporting analysis or bound for the hardware-efficient ansatze used in the experiments. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central mathematical claim—that restricting to diagonal observables is equivalent to full ANO because diagonal matrices are canonical representatives modulo unitary similarity—is presented as a direct appeal to a standard fact from linear algebra (any Hermitian operator is unitarily similar to a diagonal one). This does not reduce by construction to a self-referential definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain within the paper. The complexity reduction from O(4^k) to O(2^k) follows immediately from the restriction to diagonal entries and does not rely on smuggling an ansatz or renaming a known result. The derivation remains self-contained against external benchmarks; the variational absorption of the diagonalizing unitary is an independent modeling choice whose validity is not presupposed by the paper's equations or prior self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proposal relies on standard properties of Hermitian operators and unitary similarity from quantum information theory; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Hermitian observables are unitarily similar to diagonal matrices, making diagonals canonical representatives of the observable space.
    Invoked to justify that restricting to diagonals does not lose the capability of general ANOs.

pith-pipeline@v0.9.0 · 5709 in / 1269 out tokens · 50769 ms · 2026-05-19T15:16:43.692215+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Quantum computing in the nisq era and beyond,

    J. Preskill, “Quantum computing in the nisq era and beyond,”Quantum, vol. 2, p. 79, 2018

    work page 2018

  2. [2]

    Variational quantum algorithms,

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincioet al., “Variational quantum algorithms,”Nature Reviews Physics, vol. 3, no. 9, pp. 625– 644, 2021

    work page 2021

  3. [3]

    Noisy intermediate-scale quantum algorithms,

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menkeet al., “Noisy intermediate-scale quantum algorithms,”Reviews of Modern Physics, vol. 94, no. 1, p. 015004, 2022

    work page 2022

  4. [4]

    The theory of variational hybrid quantum-classical algorithms,

    J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The theory of variational hybrid quantum-classical algorithms,”New Journal of Physics, vol. 18, no. 2, p. 023023, 2016

    work page 2016

  5. [5]

    A variational eigenvalue solver on a photonic quantum processor,

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, “A variational eigenvalue solver on a photonic quantum processor,”Nature communications, vol. 5, no. 1, p. 4213, 2014

    work page 2014

  6. [6]

    Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,

    A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, “Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,”nature, vol. 549, no. 7671, pp. 242–246, 2017

    work page 2017

  7. [7]

    Quantum machine learning,

    J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,”Nature, vol. 549, no. 7671, pp. 195–202, 2017

    work page 2017

  8. [8]

    Effect of data encoding on the expressive power of variational quantum-machine-learning models,

    M. Schuld, R. Sweke, and J. J. Meyer, “Effect of data encoding on the expressive power of variational quantum-machine-learning models,” Physical Review A, vol. 103, no. 3, p. 032430, 2021

    work page 2021

  9. [9]

    Data re-uploading for a universal quantum classifier,

    A. P ´erez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, “Data re-uploading for a universal quantum classifier,”Quantum, vol. 4, p. 226, 2020

    work page 2020

  10. [10]

    Quantum convolutional neural networks for high energy physics data analysis,

    S. Y .-C. Chen, T.-C. Wei, C. Zhang, H. Yu, and S. Yoo, “Quantum convolutional neural networks for high energy physics data analysis,” Physical Review Research, vol. 4, no. 1, p. 013231, 2022

    work page 2022

  11. [11]

    Asynchronous training of quantum reinforcement learn- ing,

    S. Y .-C. Chen, “Asynchronous training of quantum reinforcement learn- ing,”Procedia Computer Science, vol. 222, pp. 321–330, 2023

    work page 2023

  12. [12]

    Expressibility and entan- gling capability of parameterized quantum circuits for hybrid quantum- classical algorithms,

    S. Sim, P. D. Johnson, and A. Aspuru-Guzik, “Expressibility and entan- gling capability of parameterized quantum circuits for hybrid quantum- classical algorithms,”Advanced Quantum Technologies, vol. 2, no. 12, p. 1900070, 2019

    work page 2019

  13. [13]

    Barren plateaus in quantum neural network training landscapes,

    J. R. McClean, S. Boixo, V . N. Smelyanskiy, R. Babbush, and H. Neven, “Barren plateaus in quantum neural network training landscapes,”Nature communications, vol. 9, no. 1, p. 4812, 2018

    work page 2018

  14. [14]

    An end-to- end trainable hybrid classical-quantum classifier,

    S. Y .-C. Chen, C.-M. Huang, C.-W. Hsing, and Y .-J. Kao, “An end-to- end trainable hybrid classical-quantum classifier,”Machine Learning: Science and Technology, vol. 2, no. 4, p. 045021, 2021

    work page 2021

  15. [15]

    Learning to measure quantum neural networks,

    S. Y .-C. Chen, H.-H. Tseng, H.-Y . Lin, and S. Yoo, “Learning to measure quantum neural networks,”arXiv preprint arXiv:2501.05663, 2025

    work page arXiv 2025

  16. [16]

    Adaptive non-local observable on quantum neural networks,

    H.-Y . Lin, H.-H. Tseng, S. Y .-C. Chen, and S. Yoo, “Adaptive non-local observable on quantum neural networks,” in2025 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 1. IEEE, 2025, pp. 1884–1893

    work page 2025

  17. [17]

    Quantum rein- forcement learning by adaptive non-local observables,

    H.-Y . Lin, S. Y .-C. Chen, H.-H. Tseng, and S. Yoo, “Quantum rein- forcement learning by adaptive non-local observables,” in2025 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 2. IEEE, 2025, pp. 241–246

    work page 2025

  18. [18]

    Quantum super- resolution by adaptive non-local observables,

    H.-Y . Lin, H.-H. Tseng, S. Y .-C. Chen, and S. Yoo, “Quantum super- resolution by adaptive non-local observables,” 2026, accepted at ICASSP

    work page 2026

  19. [19]

    Available: https://doi.org/10.48550/arXiv.2601.14433

    [Online]. Available: https://doi.org/10.48550/arXiv.2601.14433

    work page doi:10.48550/arxiv.2601.14433

  20. [20]

    The solovay-kitaev algorithm,

    C. M. Dawson and M. A. Nielsen, “The solovay-kitaev algorithm,” Quantum Info. Comput., vol. 6, no. 1, p. 81–95, Jan. 2006

    work page 2006

  21. [21]

    Kuperberg, Breaking the cubic barrier in the Solovay- Kitaev algorithm (2023), arXiv:2306.13158 [quant-ph]

    G. Kuperberg, “Breaking the cubic barrier in the solovay-kitaev algo- rithm,”arXiv preprint arXiv:2306.13158, 2023

    work page arXiv 2023