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REVIEW 3 major objections 4 minor 46 references

Evolution-Level Quantum Optimal Control of Single-Qubit Gates with Physics-Informed Neural Networks

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A quantum gate is better optimized as one continuous controlled evolution than as a pulse, and the paper shows the change exposes and fixes path-level structure.

desk verdict A solid representational claim with a good rotation-gate validation, but the geometric-gate diagnosis loop needs independent reference-path validation before I'd trust it. read the letter →

arxiv 2607.14884 v1 pith:UV2KNPD6 submitted 2026-07-16 quant-ph

Yao Du , Jian-Jian Cheng , Lin Zhang , Ming-Liang Hu , Xingang Wang This is my paper
classification quant-ph MSC 81Q9349K1568T07 PACS 03.67.Lx
keywords quantumoptimalcontrolphysics-informedneuralnetworkssingle-qubitgatesgeometricphaselearnabledurationBlochequationboundedprocess-leveldesign
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that single-qubit gate design should be posed not as pulse-shape optimization but as the design of a whole controlled evolution: the control fields, the Bloch-sphere trajectories, and the total duration are all optimized together as one differentiable object. Using a physics-informed neural network whose duration is a trainable time scale, it shows that rotation gates spontaneously recover the structure expected for bounded control — saturated pulses, a U-shaped duration curve matching the analytical minimum, and a channel-division law set by the rotation axis — without any pulse ansatz or duration scan. For a geometric Z gate, where correctness lives along the path, the same representation localizes the geometricity error at the intervals where the control direction turns fastest, and a turning-weighted loss reduces that error while keeping process fidelity above 0.999999. The point is that the learned object is a physical process that can be read, diagnosed, and refined, not a black-box waveform. A sympathetic reader would care because this changes what an optimizer returns: not just a high-fidelity pulse, but an inspectable account of how the gate is built in time.

What carries the argument

The central object is a physics-informed neural generator queried at physical times t_i = s_i e^τ, where τ is a trainable scalar defining the total duration T = e^τ. Its output heads produce bounded control fields and auxiliary probe/reference Bloch trajectories; a soft residual R = dr/dt − F[r, Ω, Δ] penalizes violations of the Bloch equation at sampled times, making dynamical consistency an explicit training signal rather than a post-hoc propagation. Around this core sit the gate loss (affine channel reconstruction from four probes), the time cost, and — for the geometric gate — the pointwise geometricity condition E = (1/2)h·n = 0 with h = (Ω, 0, Δ), plus a turning-rate diagnostic κ = |ΩΔ

What would settle it

Take a trained geometric Z gate, extract the learned reference trajectory, integrate the Bloch equation at high resolution under the learned fields, and measure the pointwise geometricity error E(t) and the accumulated dynamical phase directly. If the independently propagated reference path violates E(t)≈0 in the intervals where κ(t) peaks, or if the accumulated dynamical phase is not negligible, then the bottleneck diagnosis and the refinements built on it would be artifacts of the training-loss weighting rather than properties of the physical evolution.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is representational: the same physics-informed network that optimizes the gate also generates the auxiliary Bloch trajectories that define the path, and the Bloch equation is enforced as a differentiable residual along the whole interval. For rotation gates with only σx and σz controls, the optimizer recovers the expected bounded-control organization — near-saturated square pulses, durations following the analytical time-optimal reference to within a few percent, and |Ω|max/|Δ|max ≈ cot α across the axis family — although none of this structure was supplied. For the geometric Z gate, the network additionally learns two reference loops that must stay an

Load-bearing premise

The whole argument rests on the assumption that minimizing a soft Bloch-equation residual makes the network-generated trajectories — especially the reference loops that carry the geometric phase — faithful to the real evolution along the whole path; the final channel is independently checked, but the reference trajectory's physicality and the vanishing dynamical phase are never independently measured.

Editorial extensions

If this is right

  • Learned durations for rotation gates follow the analytical minimum-time curve without an external scan, so time-optimal-like controls can emerge from the process objective itself.
  • Smoothing or envelope constraints cost extra duration but leave the channel-division law intact, which implies hardware-friendly pulses can be co-designed with time and channel balance in one run.
  • The geometric Z gate reaches process fidelity above 0.999999 with pointwise geometricity enforced, so path-level correctness can be imposed as a differentiable objective rather than checked afterward.
  • Turning-weighted refinement lowers geometricity error exactly where it concentrates, demonstrating that a trained control can be locally corrected from its own read-out diagnostics.
  • The same framework carries the diagnostics (residuals, turning rates, path errors) as differentiable quantities, so a learned evolution is inspectable and improvable rather than a fixed waveform.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the read-out-and-refine loop scales to multiqubit settings, gate calibration could become targeted: localize the fragile interval in the learned evolution and reweight it, instead of re-running global pulse optimization; the paper leaves that scaling open.
  • The agreement with the analytical time bound is suggestive but not a proof of global time optimality; a natural extension is to test anisotropic amplitude bounds or measured transfer functions and see whether the emergent duration still tracks the theoretical minimum.
  • A sharper benchmark for the bottleneck claim would be to deliberately inject a narrow disturbance, such as a bandwidth notch or slow drift, at a known location and ask whether the turning-weighted loss identifies and absorbs exactly that interval; the paper's piecewise-constant comparison is a step but not a controlled-injection test.
  • If the same diagnosis-refinement idea is applied to non-Abelian or multilevel holonomic gates, success would generalize the process-level view beyond the Abelian single-qubit case; failure there would delimit the representational benefit.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a physics-informed neural network (PINN) representation for single-qubit quantum gate design in which the control fields, Bloch-state trajectories, and total gate duration are optimized together under the Bloch equation. For a family of rotation gates, it is shown that the optimized durations reproduce a Pontryagin-based reference time scale to within a few percent, that saturated square-pulse structure and channel division emerge without an imposed pulse ansatz, and that pulse-shape regularization changes durations but not the cotangent channel-division law. For a geometric Z gate, the network additionally outputs auxiliary reference loops constrained by cycle closure, orthogonality/purity, and a pointwise geometricity condition E(t)=0; the paper diagnoses localized violations of this condition around rapid turns of the control direction and introduces a turning-weighted loss that reduces the mean geometricity error while preserving high process fidelity. The central claim is representational: quantum optimal control can be treated as the design of a differentiable physical process whose internal structure can be read out, diagnosed, and refined, rather than only as pulse-shape optimization.

Significance. If the path-level claims hold, this is a useful demonstration of process-level quantum optimal control: the rotation-gate results are convincingly benchmarked against external analytic theory, and the geometric-gate construction shows how path constraints can be embedded in a differentiable objective and used for local refinement. The paper's strengths include independent RK4 validation of the rotation-gate channels, agreement of learned durations with the external Pontryagin time scale, the preserved cotangent channel division across pulse prescriptions, and the availability of code/data. The geometric-gate high-fidelity result (Fproc>0.999999) is also notable. However, the paper's central distinctive contribution—the path-level diagnosis and refinement of a geometric process—rests on the physicality of the auxiliary reference trajectories, and that physicality is not independently validated as clearly as the final-state channel is. The work is therefore promising but needs additional validation before the representational claim can be regarded as established.

major comments (3)
  1. [§III.B / App. D / Fig. 8] The geometric-gate claims (negligible dynamical phase, bottleneck diagnosis, refinement) are path-level, but the independent validation is final-state only. Sec. II.D and App. C propagate only the probe states under RK4; the trajectory residual εtraj defined in Eq. (C1) is never reported, and no protocol is given for independently propagating the reference loops n±(t). Fig. 8(b) is captioned 'RK4-evaluated geometricity error', but the methods do not state whether E(t) is evaluated on independently propagated reference states or on the network-generated reference heads. Since E(t) in Eq. (18) and Lgeo in Eq. (19)/App. D are soft losses on auxiliary curves, small E(t) on those curves is not by itself evidence that a physical evolution under the learned fields is dynamica-free. Please either (i) state explicitly that the reference heads are included in Ldyn and report εtraj for all trajecto
  2. [§III.B.3 / Fig. 8(a) / App. D] The turning-weight refinement is selected on the same metric that is then reported. Fig. 8(a) scans β and shows mean|E| decreasing to an optimum near β≈30; no multiple-seed statistics are given for the PINN runs, so the improvement and the β=50 degradation could reflect optimization noise rather than a robust effect. The discrete baseline, by contrast, uses five restarts (App. E). Please report mean ± std over seeds for the PINN protocol, and ideally a pre-specified β or a separate validation set. In addition, Fig. 8(a,c,e) compare the PINN at β=30 with the discrete baseline at β=3; without a discrete scan over the same β range or a matched computational budget, the claim that the continuous representation 'absorbs' the correction while the discrete representation 'relocates' it is not fully quantified.
  3. [Eq. (16) / §III.A] The agreement of T* with Tdirect is a strength, but the derivation of Eq. (16) is compressed. The text says Pontryagin's principle identifies saturated extremals, but the displayed formula appears to assume that at least one channel saturates and that the control vector aligns with the desired axis. For the dissipative case actually simulated (Appendix A), the reference is derived for the closed, dissipation-free problem; the match to within a few percent is then partly a statement about the weak dissipation regime. Please make explicit the assumptions behind Eq. (16) (closed system? equal bounds? saturated extremal?) and the role of dissipation in the comparison. This is not a challenge to the rotation-gate result, but it would make the claimed 'nontrivial validation' more precise.
minor comments (4)
  1. [Eq. (7)] The displayed equation for Ldyn has an apparent typo in the norm notation: the double vertical bars appear malformed. Please check the formatting.
  2. [Appendix C] εtraj is defined but never reported. Even if the geometric-gate reference loops are handled separately, reporting εtraj for the probe trajectories would give the reader a quantitative sense of path-level consistency during training.
  3. [Fig. 8 / caption] The caption says 'RK4-evaluated geometricity error' without describing how the reference states were obtained for the RK4 evaluation. Please clarify in the caption or in Appendix C/D whether the plotted E(t) is evaluated on network-generated or independently propagated reference states.
  4. [References / Supplemental Material] The Supplemental Material [34] is referenced with a placeholder URL and the movies were not available for review. Please ensure the final version includes a working link and, if possible, static versions of the key trajectories.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: rotation-gate organization is externally benchmarked; geometric refinement is optimization feedback, with a path-validation gap that is a correctness risk, not a definitional circularity.

full rationale

The main derived outputs are checked against external, parameter-free benchmarks rather than quantities defined by the fit. Optimized rotation durations are compared with the Pontryagin saturated-control scale Tdirect(α)=φmax(cosα,sinα)/Ωmax in Eq. (16), which is not supplied to the optimizer; channel division is compared with cotα in Eq. (17). The geometricity condition E(t)=h·n/2=0 in Eq. (18) is the standard definition of the local geometric condition; Lgeo in Eq. (19)/Appendix D enforces this same condition, so reporting that it becomes small after optimization is reporting constraint satisfaction, not a derived prediction. The turning-aware loop weights intervals with w=1+βκ using the same E(t) it then reports reduced; this is an optimization feedback and a β-selection artifact, not a circular derivation. The paper itself flags soft-constraint sensitivity (Sec. IV: 'because the dynamics are imposed through soft equation errors... remains a practical issue'), and the load-bearing gap is that the reference-loop physicality and 'negligible dynamical phase' are supported only by soft residuals; Fig. 8's 'RK4-evaluated geometricity error' lacks a stated protocol for independently propagating reference states, and εtraj in Eq. (C1) is not reported. That is a validation/correctness concern, not a definitional reduction. No load-bearing self-citation or imported uniqueness theorem appears.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; the turning-weight w(t) is an algorithmic construct, not a physical object. The main free inputs are loss weights and the β scan, which affect the reported quantitative results. The axioms are standard quantum-control modeling assumptions plus the soft-constraint fidelity assumption that is only partially validated.

free parameters (4)
  • β (turning-weight strength) = β≈30 (PINN); β=3 (discrete baseline)
    Scanned 0–50 to minimize mean |E| on the same trajectories used for evaluation; optimum is selected post hoc, no held-out validation.
  • Loss-balance weights (λdyn, βgate, λT, λgeo, λcyc, λamp, ζsmooth, ηbnd) = listed per case in Appendices B–D (e.g., λdyn=80, λgate=50, λgeo=10→300)
    Chosen by hand to balance dynamical, gate, time, and path terms; the geometric-gate result depends on this balance and a staged schedule.
  • Initial duration T0 = 0.7 (rotation gates)
    Initial value for the learnable duration; final T is optimized, so this mainly affects convergence, not the reported T*.
  • Network/training hyperparameters (width, layers, Ns, learning rates, steps) = width 96/256, 3/6 layers, Ns=201, lr 1e-3/3e-4, 3000/20000–30000 steps
    Chosen by hand; standard for PINN convergence and not claimed as optimized.
assumptions (5)
  • domain assumption Bloch equation/Lindblad master equation with given rates is an accurate model for the single-qubit evolution.
    Entire optimization and validation rely on this model (Eq. 1, Appendix A).
  • standard math Pontryagin maximum principle time-optimal results for bounded spin-1/2 control (Refs. [35–38]) are correct and applicable; T_direct of Eq. (16) is the right benchmark.
    Used as external reference for learned durations; not re-derived.
  • domain assumption Closed, antipodal reference loops with pointwise E(t)=0 are sufficient to realize a geometric Z gate with negligible dynamical phase.
    Standard nonadiabatic geometric phase theory; assumed in constructing loss terms (Eq. 18, Appendix D).
  • domain assumption The auxiliary reference trajectories and probe trajectories are physical solutions of the Bloch equation under the same controls.
    Enforced only through soft residual loss; no hard guarantee. Independent RK4 check of final channel is provided, not of reference path.
  • domain assumption The neural network has sufficient capacity and the optimization converges to a good local optimum.
    Not proven; standard ML assumption.

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Cite this review

Pith. "Pith review of Evolution-Level Quantum Optimal Control of Single-Qubit Gates with Physics-Informed Neural Networks." pith.science (2026). https://pith.science/paper/UV2KNPD6

@misc{pith2026260714884,
  author       = {Pith},
  title        = {Pith review of: Evolution-Level Quantum Optimal Control of Single-Qubit Gates with Physics-Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UV2KNPD6}},
  note         = {Machine review of arXiv:2607.14884}
}
read the original abstract

Quantum gate design is often represented as pulse optimization, although the physical object that implements a gate is the full controlled evolution generated by the pulse. Here we use physics-informed neural networks to represent single-qubit gate design at this evolution level: the control fields, the Bloch-state trajectories, and the total duration are learned together under the Bloch equation. This changes the optimized object from pulse amplitudes to a differentiable physical process whose structure can be inspected and refined. For rotation gates, the optimized evolutions recover the physical organization expected for bounded single-qubit control, with no prescribed pulse ansatz or duration scan. For a geometric gate, the representation identifies localized bottlenecks in maintaining the geometric condition and turns this diagnosis into feedback, reducing the residual path error while preserving high fidelity. Thus physics-informed learning is used not only to synthesize gates, but also to make optimized quantum controls physically readable, diagnosable, and locally refinable. This process-level view may be especially useful for adapting gates to hardware-specific, task-specific, and locally varying experimental constraints.

Figures

Figures reproduced from arXiv: 2607.14884 by the authors.

Figure 1
Figure 1. FIG. 1. Discrete-pulse optimization and a PINN representation of controlled gate dynamics. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Rotation axes in the controlled [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Organization under the minimal prescription. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Pulse-shape prescriptions and the geometric channel [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Automatic coordination in the geometric [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Local geometricity error in the geometric [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Turning-aware weighting of the geometricity loss. (a) [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

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Works this paper leans on

46 extracted references · 4 linked inside Pith

  1. [1]

    Schmidt and H

    M. Schmidt and H. Lipson, Distilling free-form natural laws from experimental data, Science324, 81 (2009)

    2009

  2. [2]

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. U.S.A.113, 3932 (2016)

    2016

  3. [3]

    R. Iten, T. Metger, H. Wilming, L. del Rio, and R. Renner, Discovering physical concepts with neural networks, Phys. Rev. Lett.124, 010508 (2020)

    2020

  4. [4]

    Cranmer, A

    M. Cranmer, A. Sanchez-Gonzalez, P. Battaglia, R. Xu, K. Cranmer, D. Spergel, and S. Ho, Discovering symbolic models from deep learning with inductive biases (2020), arXiv:2006.11287

    arXiv 2020

  5. [5]

    C. Brif, R. Chakrabarti, and H. Rabitz, Control of quan- tum phenomena: past, present and future, New J. Phys. 12, 075008 (2010)

    2010

  6. [6]

    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köck- enberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Training Schrödinger’s cat: quantum optimal control, Eur. Phys. J. D69, 279 (2015)

    2015

  7. [7]

    C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Quantum optimal control in quantum technologies. strategic report on current status, visions and goals for research in Europe, EPJ Quantum Technol.9, 19 (2022)

    2022

  8. [8]

    Khaneja, T

    N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynam- ics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson.172, 296 (2005)

    2005

Show all 46 references
  1. [9]

    J. P. Palao and R. Kosloff, Optimal control theory for unitary transformations, Phys. Rev. A68, 062308 (2003)

    2003

  2. [10]

    A. P. Peirce, M. A. Dahleh, and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, nu- merical approximation, and applications, Phys. Rev. A 37, 4950 (1988)

    1988

  3. [11]

    Ramakrishna, M

    V. Ramakrishna, M. V. Salapaka, M. Dahleh, H. Rabitz, and A. Peirce, Controllability of molecular systems, Phys. Rev. A51, 960 (1995)

    1995

  4. [12]

    Caneva, T

    T. Caneva, T. Calarco, and S. Montangero, Chopped random-basis quantum optimization, Phys. Rev. A84, 022326 (2011)

    2011

  5. [13]

    Machnes, E

    S. Machnes, E. Assémat, D. Tannor, and F. K. Wilhelm, Tunable, flexible, and efficient optimization of control pulses for practical qubits, Phys. Rev. Lett.120, 150401 (2018)

    2018

  6. [14]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics- informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys.378, 686 (2019)

    2019

  7. [15]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learning, Nat. Rev. Phys.3, 422 (2021)

    2021

  8. [16]

    Cuomo, V

    S. Cuomo, V. Schiano Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli, Scientific machine learning through physics-informed neural networks: Where we are and what’s next, J. Sci. Comput.92, 88 (2022)

    2022

  9. [17]

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, Neural ordinary differential equations (2018), arXiv:1806.07366

    2018 arXiv

  10. [18]

    Rackauckas, Y

    C. Rackauckas, Y. Ma, J. Martensen, C. Warner, K. Zubov, R. Supekar, D. Skinner, A. Ramadhan, and A. Edelman, Universal differential equations for scientific machine learning (2020), arXiv:2001.04385

    2020 arXiv

  11. [19]

    Güngördü and J

    U. Güngördü and J. P. Kestner, Robust quantum gates us- ing smooth pulses and physics-informed neural networks, Phys. Rev. Res.4, 023155 (2022)

    2022

  12. [20]

    Norambuena, M

    A. Norambuena, M. Mattheakis, F. J. González, and R. Coto, Physics-informed neural networks for quantum control, Phys. Rev. Lett.132, 010801 (2024)

    2024

  13. [21]

    Lauten and M

    S. Lauten and M. Otten, Physics-informed neural net- works for gate design using quantum optimal control (2025), arXiv:2511.09463

    2025

  14. [22]

    Ullah, Y

    A. Ullah, Y. Huang, M. Yang, and P. O. Dral, Physics- informed neural networks and beyond: enforcing physical constraints in quantum dissipative dynamics, Digit. Dis- cov.3, 2052 (2024)

  15. [23]

    M. V. Berry, Quantal phase factors accompanying adia- batic changes, Proc. R. Soc. London A392, 45 (1984)

    1984

  16. [24]

    Aharonov and J

    Y. Aharonov and J. Anandan, Phase change during a cyclic quantum evolution, Phys. Rev. Lett.58, 1593 (1987)

    1987

  17. [25]

    Wilczek and A

    F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett.52, 2111 (1984)

    1984

  18. [26]

    Zanardi and M

    P. Zanardi and M. Rasetti, Holonomic quantum compu- tation, Phys. Lett. A264, 94 (1999)

    1999

  19. [27]

    Zhu and Z

    S.-L. Zhu and Z. D. Wang, Implementation of universal quantum gates based on nonadiabatic geometric phases, Phys. Rev. Lett.89, 097902 (2002)

    2002

  20. [28]

    Sjöqvist, D

    E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, Non-adiabatic holonomic quantum computation, New J. Phys.14, 103035 (2012)

    2012

  21. [29]

    Zhang, T

    J. Zhang, T. H. Kyaw, S. Filipp, L. C. Kwek, E. Sjöqvist, and D. Tong, Geometric and holonomic quantum compu- tation, Phys. Rep.1027, 1 (2023)

    2023

  22. [30]

    J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Geometric quantum computation using nuclear magnetic resonance, Nature403, 869 (2000)

    2000

  23. [31]

    G. Feng, G. Xu, and G. Long, Experimental realization of nonadiabatic holonomic quantum computation, Phys. Rev. Lett.110, 190501 (2013)

    2013

  24. [32]

    A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, Experimental re- alization of non-Abelian non-adiabatic geometric gates, Nature496, 482 (2013). 16

    2013

  25. [33]

    Nagata, K

    K. Nagata, K. Kuramitani, Y. Sekiguchi, and H. Kosaka, Universal holonomic quantum gates over geometric spin qubits with polarised microwaves, Nat. Commun.9, 3227 (2018)

    2018

  26. [34]

    See Supplemental Material at [URL will be inserted by publisher] for Movie S1, showing rotation-gate trajecto- ries synchronized with the control fields, and Movie S2, showing the optimized geometricZ-gate path dynamics and control direction

  27. [35]

    Boscain and P

    U. Boscain and P. Mason, Time minimal trajectories for a spin 1/2 particle in a magnetic field, J. Math. Phys.47, 062101 (2006)

    2006

  28. [36]

    Garon, S

    A. Garon, S. J. Glaser, and D. Sugny, Time-optimal control of SU(2) quantum operations, Phys. Rev. A88, 043422 (2013)

    2013

  29. [37]

    A. D. Boozer, Time-optimal synthesis of SU(2) transfor- mations for a spin-1/2 system, Phys. Rev. A85, 012317 (2012)

    2012

  30. [38]

    Dionis and D

    E. Dionis and D. Sugny, Time-optimal control of two-level quantum systems by piecewise constant pulses, Phys. Rev. A107, 032613 (2023)

    2023

  31. [39]

    M. H. Goerz, D. Basilewitsch, F. Gago-Encinas, M. G. Krauss, K. P. Horn, D. M. Reich, and C. P. Koch, Kro- tov: A Python implementation of Krotov’s method for quantum optimal control, SciPost Phys.7, 080 (2019)

    2019

  32. [40]

    S. Wang, Y. Teng, and P. Perdikaris, Understanding and mitigating gradient flow pathologies in physics-informed neural networks, SIAM J. Sci. Comput.43, A3055 (2021)

    2021

  33. [41]

    A. S. Krishnapriyan, A. Gholami, S. Zhe, R. M. Kirby, and M. W. Mahoney, Characterizing possible fail- ure modes in physics-informed neural networks (2021), arXiv:2109.01050

    2021 arXiv

  34. [42]

    Pinn quantum control code and data, https://github.com/Xingang-Wang/PINN-quantum- control (2026), accessed July 15, 2026

    2026

  35. [43]

    Glorot and Y

    X. Glorot and Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, inProceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, Proc. Mach. Learn. Res., Vol. 9, edited by Y. W. Teh and M. Titterington (PML...

    2010

  36. [44]

    D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, inInternational Conference on Learning Representations(2015)

    2015

  37. [45]

    M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002)

    2002

  38. [46]

    C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound- constrained optimization, ACM Trans. Math. Softw.23, 550 (1997)

    1997

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