pith. sign in

arxiv: 2606.30688 · v1 · pith:WTKRVKWQnew · submitted 2026-06-28 · 🪐 quant-ph · cs.AI· cs.LG· math-ph· math.MP

A Coherence Law for Trainability in Noisy Equivariant Quantum Neural Networks

Hassan Ugail , Newton Howard This is my paper

Pith reviewed 2026-07-01 06:44 UTC · model grok-4.3

classification 🪐 quant-ph cs.AIcs.LGmath-phmath.MP
keywords U(1)-equivariant quantum circuitsnoisy trainabilitycoherence rategradient survivallight-cone reductionopen quantum systemsRayleigh quotientbrickwork circuits
0
0 comments X

The pith

The readout-visible aligned coherence rate sets the leading training law for gradients in noisy U(1)-equivariant quantum neural networks.

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

In U(1)-equivariant brickwork circuits, causality confines gradients to a light cone in the charge sector. The readout-visible aligned coherence rate, a Rayleigh quotient of the noise generator, then governs their decay under noise. Perturbative analysis turns this rate into a leading-order training law based on accumulated noise and coherence contraction. Simulations confirm degradation follows this single variable with R²=0.979. A correlated-dephasing channel test shows no loss where the rate is near zero, unlike standard diagnostics.

Core claim

The central claim is that the readout-visible aligned coherence rate determines the leading-order training law for gradient survival under noise in these circuits. The light-cone reduction pins the noiseless gradient to the sector-restricted cone independently of qubit number. The rate quantifies contraction of off-diagonal modes visible to the readout. Open-system perturbation converts the rate into the law, and simulations validate that finite-noise effects follow one accumulated variable.

What carries the argument

The readout-visible aligned coherence rate as a Rayleigh quotient of the noise generator along the gradient-carrying mode, which converts open-system decay into a training law.

If this is right

  • Gradient survival is controlled by the aligned coherence rate rather than worst-case noise rates.
  • Specific channels with high general rates but low aligned rates leave gradients intact.
  • The light-cone reduction holds with a bound independent of total qubit number.
  • Degradation depends on a single variable from noise depth and coherence contraction.
  • Sector coherence outperforms standard channel diagnostics for predicting trainability.

Where Pith is reading between the lines

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

  • Designers might target high aligned coherence rates when choosing noise models or architectures.
  • The method could extend to other symmetry groups in equivariant quantum neural networks.
  • This rate could be estimated in advance to assess trainability without full noisy simulations.
  • It connects symmetry preservation directly to measurable open-system quantities for trainability.

Load-bearing premise

The perturbative open-system analysis converting the coherence rate into the training law is valid, along with the light-cone reduction being independent of system size.

What would settle it

Observing gradient loss in the correlated-dephasing channel where the aligned rate is near zero would falsify the predicted training law.

Figures

Figures reproduced from arXiv: 2606.30688 by Hassan Ugail, Newton Howard.

Figure 1
Figure 1. Figure 1: Backward-light-cone localisation of active gradients. Numerically computed prediction-gradient second [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Perturbative sector-coherence loss. Paired small-noise simulations at [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Finite-noise accumulated coherence collapse. Gradient degradation [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sector coherence outperforms standard noise diagnostics. Predictor comparison for the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Structured noise and readout-visible aligned coherence. Anisotropic-noise study at [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

Symmetry provides a quantum neural network structure, but on its own it does not keep the network trainable once noise is present. We ask which physical quantity decides whether the gradients of an equivariant circuit survive decoherence, and we answer with a compact training law. Working with U(1)-equivariant brickwork circuits that conserve a charge, we find that two distinct effects govern a trainable gradient. Causality fixes where the gradient can live, confining it to the backward light cone of the readout inside the active charge sector. Coherence then determines how fast it decays through the contraction of the off-diagonal sector modes that the projected readout can actually observe. We prove a light-cone reduction that pins the noiseless gradient to the sector-restricted cone with a lower bound independent of the total qubit number, and we define a readout-visible aligned coherence rate as a Rayleigh quotient of the noise generator along the gradient-carrying mode. A perturbative open-system analysis turns this rate into a leading-order training law. Density-matrix simulations then confirm that the finite-noise degradation follows a single accumulated variable built from noise depth and coherence contraction, with a coefficient of determination of 0.979. The sharpest test comes from a correlated-dephasing channel that has a large worst-case rate but a near-zero aligned rate. The law predicts no gradient loss for this channel, and none is seen. Sector coherence outperforms every standard channel diagnostic we compare it against, and the analysis identifies readout-visible sector coherence as the quantity that links equivariant architecture, open-system dynamics and noisy trainability.

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 / 2 minor

Summary. The manuscript claims that in U(1)-equivariant brickwork quantum neural networks, gradient trainability under noise is governed by a readout-visible aligned coherence rate defined as a Rayleigh quotient of the noise generator along the gradient-carrying mode. A light-cone reduction is proven to pin the noiseless gradient to the sector-restricted cone with a lower bound independent of total qubit number. A perturbative open-system analysis converts this rate into a leading-order training law. Density-matrix simulations confirm degradation follows a single accumulated variable (R²=0.979), and a correlated-dephasing channel test (high worst-case rate but near-zero aligned rate) shows no loss as predicted, with sector coherence outperforming standard diagnostics.

Significance. If the central claim holds, the work identifies readout-visible sector coherence as the quantity linking equivariant structure, open-system dynamics, and noisy trainability, providing a compact law that outperforms standard channel diagnostics. Credit is due for the N-independent light-cone reduction, the falsifiable prediction confirmed by the correlated-dephasing test, and the high R² numerical support within the tested regime. This could inform design of noise-resilient symmetric QNNs.

major comments (2)
  1. [paragraph on perturbative analysis] Paragraph on perturbative analysis: the conversion of the aligned coherence rate (Rayleigh quotient of the noise generator) into the leading-order training law relies on a perturbative open-system analysis whose validity outside the weak-noise regime is assumed but not bounded; the reported R²=0.979 and zero-loss prediction for correlated dephasing are consistent with perturbation but do not rule out that the single-variable collapse is an artifact of the simulated parameter range (noise strength × depth).
  2. [light-cone reduction paragraph] Light-cone reduction paragraph: while stated as proven with an N-independent lower bound on the noiseless gradient in the sector-restricted cone, the manuscript provides no explicit derivation steps or bound expression, leaving the independence from total qubit number unverified for the gradient-carrying mode that the projected readout observes.
minor comments (2)
  1. Terminology for 'readout-visible aligned coherence rate' versus 'sector coherence' is used interchangeably in the abstract and main text; a single consistent definition would aid readability.
  2. The simulations section reports R²=0.979 but does not specify the exact noise models, depth ranges, or number of random instances used to obtain the fit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and positive assessment of the work. We address each major comment below.

read point-by-point responses

  1. Referee: the conversion of the aligned coherence rate (Rayleigh quotient of the noise generator) into the leading-order training law relies on a perturbative open-system analysis whose validity outside the weak-noise regime is assumed but not bounded; the reported R²=0.979 and zero-loss prediction for correlated dephasing are consistent with perturbation but do not rule out that the single-variable collapse is an artifact of the simulated parameter range (noise strength × depth).

    Authors: We agree that the perturbative analysis assumes the weak-noise regime and that the reported R² and special-channel result are consistent with this regime but do not exclude higher-order corrections at stronger noise. In the revision we will add an explicit discussion of the validity range of the leading-order training law, state the conditions under which higher-order terms become appreciable, and include additional density-matrix simulations at higher noise strengths to test the breakdown of the single-variable collapse. revision: yes

  2. Referee: while stated as proven with an N-independent lower bound on the noiseless gradient in the sector-restricted cone, the manuscript provides no explicit derivation steps or bound expression, leaving the independence from total qubit number unverified for the gradient-carrying mode that the projected readout observes.

    Authors: The light-cone reduction and the claimed N-independent lower bound are derived in the manuscript, yet we acknowledge that the explicit steps and the bound expression were not presented with sufficient detail. In the revised version we will supply the complete derivation of the light-cone reduction, explicitly obtain the lower bound on the noiseless gradient within the sector-restricted cone, and verify that the bound remains independent of total qubit number for the gradient-carrying mode observed by the projected readout. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent perturbative analysis and external simulation checks

full rationale

The paper defines the aligned coherence rate via Rayleigh quotient on the noise generator, proves a light-cone reduction separately, then applies perturbative open-system analysis to convert the rate into a leading-order training law. Simulations (R²=0.979) and the correlated-dephasing channel test (where aligned rate is near zero but worst-case rate is large, correctly predicting no loss) serve as independent validation rather than tautological confirmation. No self-citation chains, fitted inputs renamed as predictions, or reductions by construction appear in the load-bearing steps. The central claim remains externally falsifiable via the channel test and does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the newly defined coherence rate and two structural assumptions about the circuit and the noise treatment; no free parameters are explicitly fitted beyond the reported simulation coefficient of determination.

axioms (2)
  • domain assumption U(1)-equivariant brickwork circuits conserve charge and permit a light-cone reduction that confines the noiseless gradient to the sector-restricted cone independently of qubit number
    Invoked to establish where the gradient can live before coherence effects act.
  • domain assumption Perturbative open-system analysis applies to convert the aligned coherence rate into a leading-order training law
    Used to obtain the compact training law from the Rayleigh quotient.
invented entities (1)
  • readout-visible aligned coherence rate no independent evidence
    purpose: Quantifies the contraction rate of off-diagonal sector modes that the projected readout can observe
    Introduced as a Rayleigh quotient of the noise generator along the gradient-carrying mode; no independent falsifiable evidence outside the paper is provided.

pith-pipeline@v0.9.1-grok · 5820 in / 1701 out tokens · 41667 ms · 2026-07-01T06:44:03.840514+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 35 canonical work pages · 3 internal anchors

  1. [1]

    Quantum machine learning,

    J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,”Nature, vol. 549, pp. 195–202, 2017.https://doi.org/10.1038/nature23474

    work page doi:10.1038/nature23474 2017

  2. [2]

    Quantum computing in the NISQ era and beyond,

    J. Preskill, “Quantum computing in the NISQ era and beyond,”Quantum, vol. 2, art. 79, 2018. https://doi. org/10.22331/q-2018-08-06-79

    work page internal anchor Pith review doi:10.22331/q-2018-08-06-79 2018

  3. [3]

    Variational quantum algorithms,

    M. Cerezoet al., “Variational quantum algorithms,”Nature Reviews Physics, vol. 3, pp. 625–644, 2021. https: //doi.org/10.1038/s42254-021-00348-9

    work page doi:10.1038/s42254-021-00348-9 2021

  4. [4]

    Quantum circuit learning,

    K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, “Quantum circuit learning,”Physical Review A, vol. 98, art. 032309, 2018.https://doi.org/10.1103/PhysRevA.98.032309

    work page doi:10.1103/physreva.98.032309 2018

  5. [5]

    Supervised learning with quantum-enhanced feature spaces,

    V . Havlíˇceket al., “Supervised learning with quantum-enhanced feature spaces,”Nature, vol. 567, pp. 209–212, 2019.https://doi.org/10.1038/s41586-019-0980-2

    work page doi:10.1038/s41586-019-0980-2 2019

  6. [6]

    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, art. 032430, 2021. https://doi.org/10.1103/ PhysRevA.103.032430

    2021

  7. [7]

    The power of quantum neural networks,

    A. Abbaset al., “The power of quantum neural networks,”Nature Computational Science, vol. 1, pp. 403–409, 2021.https://doi.org/10.1038/s43588-021-00084-1

    work page doi:10.1038/s43588-021-00084-1 2021

  8. [8]

    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, art. 4812, 2018. https://doi.org/10.1038/ s41467-018-07090-4

    2018

  9. [9]

    Cost function dependent barren plateaus in shallow parametrized quantum circuits,

    M. Cerezo, A. Sone, T. V olkoff, L. Cincio, and P. J. Coles, “Cost function dependent barren plateaus in shallow parametrized quantum circuits,”Nature Communications, vol. 12, art. 1791, 2021. https://doi.org/10.1038/ s41467-021-21728-w

    2021

  10. [10]

    Connecting ansatz expressibility to gradient magnitudes and barren plateaus,

    Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, “Connecting ansatz expressibility to gradient magnitudes and barren plateaus,”PRX Quantum, vol. 3, art. 010313, 2022. https://doi.org/10.1103/PRXQuantum.3. 010313 12 APREPRINT- JULY1, 2026

    work page doi:10.1103/prxquantum.3 2022

  11. [11]

    A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits,

    M. Ragoneet al., “A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits,”Nature Communications, vol. 15, art. 7172, 2024.https://doi.org/10.1038/s41467-024-49909-3

    work page doi:10.1038/s41467-024-49909-3 2024

  12. [12]

    Noise-induced barren plateaus in variational quantum algorithms,

    S. Wanget al., “Noise-induced barren plateaus in variational quantum algorithms,”Nature Communications, vol. 12, art. 6961, 2021.https://doi.org/10.1038/s41467-021-27045-6

    work page doi:10.1038/s41467-021-27045-6 2021

  13. [13]

    Limitations of optimization algorithms on noisy quantum devices,

    D. Stilck França and R. García-Patrón, “Limitations of optimization algorithms on noisy quantum devices,”Nature Physics, vol. 17, pp. 1221–1227, 2021.https://doi.org/10.1038/s41567-021-01356-3

    work page doi:10.1038/s41567-021-01356-3 2021

  14. [14]

    Noise resilience of variational quantum compiling,

    K. Sharma, S. Khatri, M. Cerezo, and P. J. Coles, “Noise resilience of variational quantum compiling,”New Journal of Physics, vol. 22, art. 043006, 2020.https://doi.org/10.1088/1367-2630/ab784c

    work page doi:10.1088/1367-2630/ab784c 2020

  15. [15]

    Group Equivariant Convolutional Networks

    T. Cohen and M. Welling, “Group equivariant convolutional networks,”Proceedings of the 33rd International Conference on Machine Learning, pp. 2990–2999, 2016. https://proceedings.mlr.press/v48/cohenc16. html. arXiv DOI:https://doi.org/10.48550/arXiv.1602.07576

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1602.07576 2016

  16. [16]

    Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges

    M. M. Bronstein, J. Bruna, T. Cohen, and P. Veli ˇckovi’c, “Geometric deep learning: grids, groups, graphs, geodesics, and gauges,”arXiv:2104.13478, 2021.https://doi.org/10.48550/arXiv.2104.13478

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2104.13478 2021

  17. [17]

    Group-invariant quantum machine learning,

    M. Laroccaet al., “Group-invariant quantum machine learning,”PRX Quantum, vol. 3, art. 030341, 2022. https://doi.org/10.1103/PRXQuantum.3.030341

    work page doi:10.1103/prxquantum.3.030341 2022

  18. [18]

    Theory for equivariant quantum neural networks,

    Q. T. Nguyenet al., “Theory for equivariant quantum neural networks,”PRX Quantum, vol. 5, art. 020328, 2024. https://doi.org/10.1103/PRXQuantum.5.020328

    work page doi:10.1103/prxquantum.5.020328 2024

  19. [19]

    Theoretical guarantees for permutation- equivariant quantum neural networks,

    L. Schätzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, “Theoretical guarantees for permutation- equivariant quantum neural networks,”npj Quantum Information, vol. 10, art. 12, 2024. https://doi.org/10. 1038/s41534-024-00804-1

    2024

  20. [20]

    Exploiting symmetry in variational quantum machine learning,

    J. J. Meyeret al., “Exploiting symmetry in variational quantum machine learning,”PRX Quantum, vol. 4, art. 010328, 2023.https://doi.org/10.1103/PRXQuantum.4.010328

    work page doi:10.1103/prxquantum.4.010328 2023

  21. [21]

    Symmetry-organised complexity in quantum neural networks,

    H. Ugail and N. Howard, “Symmetry-organised complexity in quantum neural networks,”Symmetry, vol. 18, no. 6, art. 912, 2026.https://doi.org/10.3390/sym18060912

    work page doi:10.3390/sym18060912 2026

  22. [22]

    Absence of barren plateaus in quantum convolutional neural networks,

    A. Pesahet al., “Absence of barren plateaus in quantum convolutional neural networks,”Physical Review X, vol. 11, art. 041011, 2021.https://doi.org/10.1103/PhysRevX.11.041011

    work page doi:10.1103/physrevx.11.041011 2021

  23. [23]

    Generalization in quantum machine learning from few training data,

    M. C. Caroet al., “Generalization in quantum machine learning from few training data,”Nature Communications, vol. 13, art. 4919, 2022.https://doi.org/10.1038/s41467-022-32550-3

    work page doi:10.1038/s41467-022-32550-3 2022

  24. [24]

    Quantum machine learning beyond kernel methods,

    S. Jerbiet al., “Quantum machine learning beyond kernel methods,”Nature Communications, vol. 14, art. 517, 2023.https://doi.org/10.1038/s41467-023-36159-y

    work page doi:10.1038/s41467-023-36159-y 2023

  25. [25]

    Evaluation of latent diffusion enhanced face recognition under forensic image degradations,

    H. Ugail, H. M. Alawar, A. A. Zehi, A. M. Alkendi, and I. L. Jaleel, “Evaluation of latent diffusion enhanced face recognition under forensic image degradations,”Discover Computing, vol. 29, art. 193, 2026. https: //doi.org/10.1007/s10791-026-10082-4

    work page doi:10.1007/s10791-026-10082-4 2026

  26. [26]

    Deep face recognition using imperfect facial data,

    A. Elmahmudi and H. Ugail, “Deep face recognition using imperfect facial data,”Future Generation Computer Systems, vol. 99, pp. 213–225, 2019.https://doi.org/10.1016/j.future.2019.04.025

    work page doi:10.1016/j.future.2019.04.025 2019

  27. [27]

    Deep transfer learning for visual analysis and attribution of paintings by Raphael,

    H. Ugail, D. G. Stork, H. G. M. Edwards, S. C. Seward, and C. Brooke, “Deep transfer learning for visual analysis and attribution of paintings by Raphael,”Heritage Science, vol. 11, article 268, 2023. https://doi.org/10. 1186/s40494-023-01094-0

    2023

  28. [28]

    Is facial beauty in the eyes? A multi-method approach to interpreting facial beauty prediction in machine learning models,

    A. A. Ibrahim, N. H. Ugail, and H. Ugail, “Is facial beauty in the eyes? A multi-method approach to interpreting facial beauty prediction in machine learning models,”Discover Artificial Intelligence, vol. 5, p. 16, 2025. https://doi.org/10.1007/s44163-025-00226-8

    work page doi:10.1007/s44163-025-00226-8 2025

  29. [29]

    Quantifying the dynamics of consciousness using hierarchical integration, organ- ised complexity and metastability,

    H. Ugail and N. Howard, “Quantifying the dynamics of consciousness using hierarchical integration, organ- ised complexity and metastability,”arXiv:2512.10972, Dec. 2025. https://doi.org/10.48550/arXiv.2512. 10972

    work page doi:10.48550/arxiv.2512 2025

  30. [30]

    Extending Noether’s theorem by quantifying the asymmetry of quantum states,

    I. Marvian and R. W. Spekkens, “Extending Noether’s theorem by quantifying the asymmetry of quantum states,” Nature Communications, vol. 5, art. 3821, 2014.https://doi.org/10.1038/ncomms4821

    work page doi:10.1038/ncomms4821 2014

  31. [31]

    Robustness of asymmetry and coherence of quantum states,

    M. Pianiet al., “Robustness of asymmetry and coherence of quantum states,”Physical Review A, vol. 93, art. 042107, 2016.https://doi.org/10.1103/PhysRevA.93.042107

    work page doi:10.1103/physreva.93.042107 2016

  32. [32]

    Quantifying coherence,

    T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,”Physical Review Letters, vol. 113, art. 140401, 2014.https://doi.org/10.1103/PhysRevLett.113.140401

    work page doi:10.1103/physrevlett.113.140401 2014

  33. [33]

    A channel-level diagnostic for symmetry breaking in noisy equivariant quantum neural networks,

    H. Ugail and N. Howard, “A channel-level diagnostic for symmetry breaking in noisy equivariant quantum neural networks,”IEEE Access, vol. 14, 2026.https://doi.org/10.1109/ACCESS.2026.3706394 13 APREPRINT- JULY1, 2026

    work page doi:10.1109/access.2026.3706394 2026

  34. [34]

    On the generators of quantum dynamical semigroups,

    G. Lindblad, “On the generators of quantum dynamical semigroups,”Communications in Mathematical Physics, vol. 48, pp. 119–130, 1976.https://doi.org/10.1007/BF01608499

    work page doi:10.1007/bf01608499 1976

  35. [35]

    Completely positive dynamical semigroups of N-level systems,

    V . Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,”Journal of Mathematical Physics, vol. 17, pp. 821–825, 1976.https://doi.org/10.1063/1.522979

    work page doi:10.1063/1.522979 1976

  36. [36]

    Campa, T

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems. Oxford University Press, Oxford, 2002. https://doi.org/10.1093/acprof:oso/9780199213900.001.0001

    work page doi:10.1093/acprof:oso/9780199213900.001.0001 2002

  37. [37]

    An initialization strategy for addressing barren plateaus in parametrized quantum circuits,

    E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, “An initialization strategy for addressing barren plateaus in parametrized quantum circuits,”Quantum, vol. 3, art. 214, 2019. https://doi.org/10.22331/ q-2019-12-09-214

    2019

  38. [38]

    Evaluating analytic gradients on quantum hardware,

    M. Schuld, V . Bergholm, C. Gogolin, J. Izaac, and N. Killoran, “Evaluating analytic gradients on quantum hardware,”Physical Review A, vol. 99, art. 032331, 2019. https://doi.org/10.1103/PhysRevA.99.032331

    work page doi:10.1103/physreva.99.032331 2019

  39. [39]

    General parameter-shift rules for quantum gradients,

    D. Wierichs, J. Izaac, C. Wang, and C. Y .-Y . Lin, “General parameter-shift rules for quantum gradients,”Quantum, vol. 6, art. 677, 2022.https://doi.org/10.22331/q-2022-03-30-677

    work page doi:10.22331/q-2022-03-30-677 2022

  40. [40]

    Covariant quantum Markovian evolutions,

    A. S. Holevo, “Covariant quantum Markovian evolutions,”Journal of Mathematical Physics, vol. 37, pp. 1812– 1832, 1996.https://doi.org/10.1063/1.531481

    work page doi:10.1063/1.531481 1996

  41. [41]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2010.https://doi.org/10.1017/CBO9780511976667 14

    work page doi:10.1017/cbo9780511976667 2010