arxiv: 2604.11314 · v2 · submitted 2026-04-13 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Fidelity-informed neural pulse compilation of a continuous family of quantum gates with uncertainty-margin analysis

Arash Fath Lipaei , Ebrahim Khaleghian , Selin Aslan , Gani G\"oral , Zidong Lin , \"Ozg\"ur E. M\"ustecapl{\i}o\u{g}lu

Authors on Pith no claims yet

Pith reviewed 2026-05-15 07:32 UTC · model grok-4.3

classification 🪐 quant-ph

keywords neural pulse compilationquantum gate synthesisNMR quantum processorunitary fidelityuncertainty margin analysiscontinuous gate familyrisk-aware optimizationSU(2) gate compilation

0 comments

The pith

A neural network learns to map any single-qubit gate parameters directly to radio-frequency pulse sequences on an NMR processor.

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

The paper develops a neural framework that takes the axis-angle parameters of an arbitrary single-qubit gate in SU(2) and outputs a piecewise-constant radio-frequency control sequence that realizes the gate on a three-qubit liquid-state NMR device. Training occurs end-to-end by computing the time-ordered propagator of the driven Hamiltonian and using global-phase-insensitive unitary fidelity as the learning signal, so that a single model covers the entire continuous family of gates. Numerical experiments confirm generalization across parameter values, and benchtop hardware tests validate representative compiled pulses. A separate risk-aware stage using right-tail Conditional Value-at-Risk redesigns the pulses to enlarge their tolerance margins against structured uncertainties in Hamiltonian and control parameters.

Core claim

A single neural model, trained end-to-end through the time-ordered propagator of the driven Hamiltonian with unitary fidelity as the objective, produces piecewise-constant radio-frequency pulses that implement any axis-angle parameterized single-qubit unitary on a three-qubit NMR processor; the same model generalizes across the continuous parameter range, achieves experimental validation on representative cases, and yields broader uncertainty margins after a Conditional Value-at-Risk redesign.

What carries the argument

The neural pulse compiler that maps SU(2) axis-angle parameters to piecewise-constant radio-frequency control sequences, trained by back-propagation through the time-ordered exponential of the driven Hamiltonian using phase-insensitive fidelity.

Load-bearing premise

The driven Hamiltonian model used to compute the propagator during training accurately represents the real experimental system and the chosen uncertainty set captures the dominant structured perturbations.

What would settle it

Applying the compiled pulses to the physical NMR device across a dense sampling of gate parameters and observing actual fidelities that fall well below the numerically predicted values for inputs outside the training distribution would falsify the generalization claim.

Figures

Figures reproduced from arXiv: 2604.11314 by Arash Fath Lipaei, Ebrahim Khaleghian, Gani G\"oral, \"Ozg\"ur E. M\"ustecapl{\i}o\u{g}lu, Selin Aslan, Zidong Lin.

**Figure 1.** Figure 1: Distribution of fidelities on the 3◦ mesh over (γ, θ, ϕ) within [0, 90◦ ] 3 in the nominal setting. where t is the (1 − α) quantile of the loss distribution. In the present paper, RU-CVaR is interpreted as a design principle that penalizes fragile pulse solutions by emphasizing adverse realizations within the chosen uncertainty set. It is therefore used to improve tolerance margins under prescribed perturb… view at source ↗

**Figure 2.** Figure 2: Mean fidelity versus perturbation level for selected error channels. The comparison il [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Output spectrum of the NMR device after applying a short square pulse to the thermal [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Tomography result for the pseudo-pure state (PPS), with a reconstructed-state fidelity of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Tomography result for the density matrix after applying the compiled one-qubit gate to [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Output NMR spectra after applying the one-qubit gate to the PPS for different values of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Phase between the real and imaginary parts of the output NMR spectrum versus [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: Noiseless training. C.1 Goal and Setup We re-optimize the noiselessly pretrained network against sampled perturbation scenarios. The approach is scenario-based risk minimization [25, 44, 13, 26]: for each training example (target gate), we draw S uncertainty scenarios, propagate the system under each scenario, evaluate the unitary mismatch via fidelity, and optimize a risk-averse aggregate loss (RU-CVaR) c… view at source ↗

**Figure 9.** Figure 9: Stochastic training. D Uncertainty-margin results This appendix collects the detailed perturbation sweeps for nominally trained and risk-aware controllers. Each panel shows the average fidelity over randomly sampled target gates while sweeping a single perturbation channel and keeping the remaining parameters fixed at their nominal values; see figure 10. The purpose of these plots is to visualize sensitivi… view at source ↗

**Figure 10.** Figure 10: Mean fidelity versus errors 16 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗

read the original abstract

We develop a fidelity-informed neural pulse-compilation framework for a continuous family of single-qubit gates on a three-qubit liquid-state nuclear magnetic resonance (NMR) processor. Instead of decomposing each target unitary into a sequence of calibrated basis gates, the method learns a direct map from the axis-angle parameters of an arbitrary U_2 in SU(2) operation to a piecewise-constant radio-frequency control sequence that implements the desired transformation. Training is performed end-to-end through the time-ordered propagator of the driven Hamiltonian using global-phase-insensitive unitary fidelity as the learning signal. We show numerically that a single model generalizes across a continuous range of gate parameters and experimentally validate representative compiled pulses on a benchtop three-qubit NMR device. In addition, we analyze sensitivity to structured perturbations in Hamiltonian and control parameters by introducing a prescribed uncertainty set and performing a comparative risk-aware redesign based on right-tail Conditional Value-at-Risk (RU-CVaR). This stage produces pulse solutions with broader tolerance margins within the chosen uncertainty model. The results demonstrate continuous pulse-level gate synthesis in an experimentally accessible setting and illustrate a hardware-aware compilation strategy that can be extended to other quantum platforms. While the uncertainty model considered here is tailored to NMR, the neural compilation and risk-aware optimization framework are general and may be useful in architectures where calibration overhead, parameter drift, or control constraints make repeated per-gate optimization costly.

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

A neural compiler maps continuous SU(2) parameters straight to NMR pulses with end-to-end fidelity training and adds a CVaR redesign for parameter tolerance; the experimental spot-checks are the main concrete advance.

read the letter

The paper trains one neural net to output piecewise-constant RF sequences for any single-qubit gate given its axis-angle parameters, using the time-ordered propagator and global-phase-insensitive fidelity as the loss. It reports numerical generalization across the continuous family and runs a few compiled pulses on a benchtop three-qubit NMR device. A second stage redesigns the pulses inside a prescribed uncertainty set using right-tail CVaR to widen tolerance margins to Hamiltonian and control drifts. That combination is the actual new piece: direct parameter-to-pulse mapping plus an explicit risk-aware step on real hardware, rather than another decomposition routine. The end-to-end training avoids the usual calibrated basis-gate overhead, and the experimental comparisons give direct fidelity numbers for the tested points, which is useful even if limited to representatives. The uncertainty redesign is straightforward to implement and produces visibly broader margins within the chosen set. The main limitation is that everything is tuned to the NMR driven-Hamiltonian model and the specific uncertainty bounds they picked; the gains are shown only inside that model, so transfer to other platforms would require re-deriving the set and re-training. The numerical results are also generated under the exact simulator used for training, which is normal but leaves open how much model mismatch would degrade performance on a different device. No large statistical tests on the robustness improvement are presented, but the comparative redesign is shown clearly enough. This is for groups doing pulse-level control on NMR or similar analog systems where repeated per-gate optimization is expensive. The method is concrete, the hardware data exist, and the risk step is a practical addition, so it deserves a serious referee even if the claims stay incremental.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a fidelity-informed neural pulse-compilation framework that learns a direct map from the axis-angle parameters of an arbitrary single-qubit gate in SU(2) to a piecewise-constant RF control sequence on a three-qubit liquid-state NMR processor. Training occurs end-to-end through the time-ordered propagator of the driven Hamiltonian with global-phase-insensitive unitary fidelity as the objective. Numerical results demonstrate that a single model generalizes across a continuous range of gate parameters; representative compiled pulses are validated experimentally on a benchtop NMR device; and a comparative risk-aware redesign using right-tail Conditional Value-at-Risk (RU-CVaR) within a prescribed uncertainty set is shown to produce solutions with broader tolerance margins.

Significance. If the reported numerical generalization and experimental fidelity comparisons hold, the work supplies a concrete, hardware-aware demonstration of continuous pulse-level gate synthesis that bypasses per-gate basis decomposition and calibration. The explicit incorporation of an uncertainty set and RU-CVaR redesign constitutes a reproducible robustness analysis that directly addresses parameter drift and control constraints; these elements are strengths that could be extended to other platforms where repeated optimization is costly.

minor comments (3)

[Abstract] Abstract: the claim of numerical generalization and experimental validation would be strengthened by including at least one quantitative fidelity value (e.g., average or worst-case fidelity) and a brief statement of the training set size or architecture depth.
[Experimental validation] The experimental section should explicitly state the number of distinct gate parameters tested on the benchtop device and report the corresponding simulated-versus-measured fidelity pairs in a table to allow direct assessment of model-to-hardware transfer.
[Uncertainty analysis] The uncertainty-set bounds and the precise definition of the RU-CVaR objective (including the tail probability level) should be given in a dedicated equation or table so that the redesign step can be reproduced without ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The summary accurately captures the core contributions of the fidelity-informed neural pulse-compilation framework, its generalization across continuous gate families, experimental validation on the NMR device, and the RU-CVaR-based robustness analysis. We appreciate the recognition that these elements provide a hardware-aware demonstration applicable to other platforms.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central pipeline trains a neural network end-to-end on unitary fidelity computed from the driven Hamiltonian propagator; this is a standard supervised-learning construction with no reduction of the reported generalization or experimental fidelity to a fitted parameter by definition. Numerical results are obtained under the identical forward model used for training, and experimental spot-checks are direct comparisons on the benchtop device. The RU-CVaR redesign operates inside an explicitly prescribed uncertainty set as an additional risk-aware optimization step rather than a self-referential fit. No load-bearing self-citation, ansatz smuggling, or uniqueness theorem imported from the authors' prior work appears in the derivation chain. The method therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the simulated time-ordered propagator matches experiment closely enough for the learned mapping to transfer, plus the modeling choice that the prescribed uncertainty set adequately represents real parameter drift; no new physical entities are postulated.

free parameters (2)

neural-network weights and architecture
Learned during end-to-end training on fidelity; their specific values determine the compiled pulses.
uncertainty-set bounds
Chosen to define the structured perturbations for the RU-CVaR redesign step.

axioms (1)

domain assumption The driven Hamiltonian accurately describes the NMR spin dynamics under piecewise-constant RF control
Invoked when computing the time-ordered propagator used as the forward model for training.

pith-pipeline@v0.9.0 · 5587 in / 1370 out tokens · 39622 ms · 2026-05-15T07:32:26.300166+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Training is performed end-to-end through the time-ordered propagator of the driven Hamiltonian using global-phase-insensitive unitary fidelity as the learning signal.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

comparative risk-aware redesign based on right-tail Conditional Value-at-Risk (RU-CVaR)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

[1]

C P Koch, U Boscain, T Calarco, G Dirr, S Filipp, S J Glaser, R Kosloff, S Montangero, T Schulte-Herbr¨ uggen, D Sugny and F K Wilhelm 2022 Quantum optimal control in quantum technologies: Strategic report on current status, visions and goals for research in EuropeEPJ Quantum Technol.919

work page 2022
[2]

Commun.124989

L Clinton, J Bausch and T Cubitt 2021 Hamiltonian simulation algorithms for near-term quantum hardwareNat. Commun.124989

work page 2021
[3]

L M K Vandersypen and I L Chuang 2005 NMR techniques for quantum control and computationRev. Mod. Phys.761037–1069

work page 2005
[4]

N A Gershenfeld and I L Chuang 1997 Bulk spin-resonance quantum computationScience 275350–356

work page 1997
[5]

Natl Acad

D G Cory, A F Fahmy and T F Havel 1997 Ensemble quantum computing by NMR spectroscopyProc. Natl Acad. Sci. USA941634–1639

work page 1997
[6]

N Khaneja, T Reiss, C Kehlet, T Schulte-Herbr¨ uggen and S J Glaser 2005 Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithmsJ. Magn. Reson.172296–305

work page 2005
[7]

A Barenco, C H Bennett, R Cleve, D P DiVincenzo, N Margolus, P Shor, T Sleator, J A Smolin and H Weinfurter 1995 Elementary gates for quantum computationPhys. Rev. A52 3457–3467

work page 1995
[8]

Comput.6 81–95

C M Dawson and M A Nielsen 2005 The Solovay–Kitaev algorithmQuantum Inf. Comput.6 81–95

work page 2005
[9]

M Cerezo, A Arrasmith, R Babbush, S C Benjamin, S Endo, K Fujii, J R McClean, K Mitarai, X Yuan, L Cincio and P J Coles 2021 Variational quantum algorithmsNat. Rev. Phys.3 625–644

work page 2021
[10]

A Kandala, A Mezzacapo, K Temme, M Takita, M Brink, J M Chow and J M Gambetta 2017 Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets Nature549242–246

work page 2017
[11]

Commun.103007

H R Grimsley, S E Economou, E Barnes and N J Mayhall 2019 An adaptive variational algorithm for exact molecular simulations on a quantum computerNat. Commun.103007

work page 2019
[12]

Quantum Eng.31–13

M Ibrahim, H Mohammadbagherpoor, C Rios, N T Bronn and G T Byrd 2022 Pulse-level optimization of parameterized quantum circuits for variational quantum algorithmsIEEE Trans. Quantum Eng.31–13

work page 2022
[13]

C Acerbi and D Tasche 2002 On the coherence of expected shortfallJ. Bank. Finance26 1487–1503

work page 2002
[14]

T-S Ho, J Dominy and H Rabitz 2006 Why do effective quantum controls appear easy to find? Phys. Rev. A79013422

work page 2006
[15]

R Chakrabarti and H Rabitz 2007 Quantum control landscapesInt. Rev. Phys. Chem.26 671–735

work page 2007
[16]

M H Levitt 1986 Composite pulsesProg. Nucl. Magn. Reson. Spectrosc.1861–122

work page 1986
[17]

Z-C Shi, J-T Ding, Y-H Chen, J Song, Y Xia, X X Yi and F Nori 2024 Supervised learning for robust quantum control in composite-pulse systemsPhys. Rev. Applied21044012

work page 2024
[18]

Commun.115396

T Proctor, K Rudinger, K Young, M Sarovar and R Blume-Kohout 2020 Detecting and tracking drift in quantum information processorsNat. Commun.115396

work page 2020
[19]

J Kelly, R Barends, A G Fowler, A Megrant, E Jeffrey, T C White, D Sank, J Y Mutus, B Campbell, Y Chen, Z Chen, B Chiaro, A Dunsworth, C Neill, P J J O’Malley, C Quintana, P Roushan, A Vainsencher, J Wenner, A N Korotkov, A N Cleland and J M Martinis 2016 Scalable in situ qubit calibration during repetitive error detectionPhys. Rev. A94032321 17

work page 2016
[20]

G A L White, C Hill, A Polloreno, C D Hill, L C L Hollenberg, S J Pauka and A Morello 2021 Performance optimization for drift-robust fidelity in singlet-triplet qubitsPhys. Rev. Applied 15014023

work page 2021
[21]

S J Evered, D Bluvstein, M Kalinowski, S Ebadi, A Omran, H Levine, T Graß, T T Wang, R Samajdar, X Wang, N Maskara, J D Whalen, M D Lukin, H Pichler, S Choi, M Greiner and V Vuleti´ c 2023 High-fidelity parallel entangling gates on a neutral-atom quantum computer Nature622268–272

work page 2023
[22]

D Bluvstein, S H Li, H Pichler, M Kalinowski, S Ebadi, S J Evered, A Omran, N Maskara, A Semeghini, M Greiner and M D Lukin 2024 Logical quantum processor based on reconfigurable atom arraysNature62658–65

work page 2024
[23]

F Sauvage and F Mintert 2022 Optimal control of families of quantum gatesPhys. Rev. Lett. 129050507

work page 2022
[24]

Process.25131

E Khaleghian, A F Lipaei, A Bahrampouret al2026 Development of neural network-based optimal control pulse generator for quantum logic gates using the GRAPE algorithm in NMR quantum computerQuantum Inf. Process.25131

work page
[25]

Risk221–41

R T Rockafellar and S Uryasev 2000 Optimization of conditional value-at-riskJ. Risk221–41

work page 2000
[26]

A Shapiro, D Dentcheva and A Ruszczy´ nski 2014Lectures on Stochastic Programming: Modeling and Theory(Philadelphia: SIAM)

work page
[27]

Rep.51–15

M Tamuraet al2021 Review of the temporal stability of the magnetic field for ultra-high-field MRI magnetsNIMS Tech. Rep.51–15

work page
[28]

Rep.624564

S-Y Tsaiet al2016 Effects of frequency drift on the quantification of gamma-aminobutyric acid using MEGA-PRESSSci. Rep.624564

work page
[29]

Cryogenic Ltd 2019 Magnetic resonance magnet systems: technical brochure available at https://www.cryogenic.co.uk/(accessed 13 Nov 2025)

work page 2019
[30]

H Ryan, A Smith and M Utz 2014 Structural shimming for high-resolution nuclear magnetic resonance spectroscopy in lab-on-a-chip devicesLab Chip141678–1685

work page 2014
[31]

Q Teng 2005 Instrumentation and practical aspectsStructural Biology: Practical NMR Applications(Berlin: Springer) pp 57–148

work page 2005
[32]

R K Harriset al2008 Further conventions for NMR shielding and chemical shiftsAnn. Magn. Reson.71–31

work page
[33]

K Trainoret al2019 Temperature dependence of NMR chemical shiftsProg. Nucl. Magn. Reson. Spectrosc.1101–29

work page
[34]

J De Poorter 1995 Noninvasive MRI thermometry with the proton resonance frequency methodJ. Magn. Reson. Imaging6157–163

work page 1995
[35]

V Rieke and K B Pauly 2008 MRI thermometryJ. Magn. Reson. Imaging27376–390

work page 2008
[36]

H O Reich 2016 Spin–spin splitting and J-coupling lecture notes available at https://organicchemistrydata.org(accessed 13 Nov 2025)

work page 2016
[37]

Commun.151234

D Torodiiet al2024 1H J-couplings in fast magic-angle-spinning solid-state NMRNat. Commun.151234

work page
[38]

Phys.108583–595

A L Estebanet al2010 Vibrational contributions to vicinal proton–proton coupling constants 3JHH Mol. Phys.108583–595

work page
[39]

J C Schug, P E McMahon and H S Gutowsky 1960 Electron coupling of nuclear spins IV: temperature dependence in substituted ethanesJ. Chem. Phys.331400–1407

work page 1960
[40]

Teledyne LeCroy 2024 T3AWG6K series arbitrary waveform generators data sheet revision 1.0

work page 2024
[41]

Siglent Technologies 2022 SDG7000A series arbitrary waveform generators service manual version EN01A 18

work page 2022
[42]

Zurich Instruments 2019 HDAWG: high definition arbitrary waveform generator product brochure

work page 2019
[43]

Rigol Technologies 2021 DG70000 arbitrary waveform generator data sheet

work page 2021
[44]

R T Rockafellar and S Uryasev 2002 Conditional value-at-risk for general loss distributionsJ. Bank. Finance261443–1471

work page 2002
[45]

M Hamermesh 1962Group Theory and Its Application to Physical Problems(Reading, Massachusetts: Addison-Wesley)

work page
[46]

H Singh, Arvind and K Dorai 2016 Constructing valid density matrices on an NMR quantum information processor via maximum likelihood estimationPhys. Lett. A3803051–3056

work page 2016
[47]

A Gaikwad, G Singh, R Bhole, V Singh and A Kumar 2018 Experimental demonstration of selective quantum process tomography without ancilla in a three-qubit NMR systemPhys. Rev. A97022311

work page 2018
[48]

Optimal control design of constant amplitude phase-modulated pulses: Application to calibration-free broadband excitation,

T. E. Skinner, K. Kobzar, B. Luy, M. R. Bendall, W. Bermel, N. Khaneja, and S. J. Glaser, “Optimal control design of constant amplitude phase-modulated pulses: Application to calibration-free broadband excitation,”J. Magn. Reson.179, 241–249 (2006). doi:10.1016/j.jmr.2005.12.010

work page doi:10.1016/j.jmr.2005.12.010 2006
[49]

Reducing the duration of broadband excitation pulses using optimal control with limited RF amplitude,

T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja, and S. J. Glaser, “Reducing the duration of broadband excitation pulses using optimal control with limited RF amplitude,”J. Magn. Reson.167, 68–74 (2004). doi:10.1016/j.jmr.2003.12.001

work page doi:10.1016/j.jmr.2003.12.001 2004
[50]

Exploring the limits of broadband excitation and inversion: II. RF-power optimized pulses,

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser, and B. Luy, “Exploring the limits of broadband excitation and inversion: II. RF-power optimized pulses,”J. Magn. Reson.194, 58–66 (2008). doi:10.1016/j.jmr.2008.05.023

work page doi:10.1016/j.jmr.2008.05.023 2008
[51]

Tunable, Flexible, and Efficient Optimization of Control Pulses for Practical Qubits,

S. Machnes, E. Ass´ emat, D. Tannor, and F. K. Wilhelm, “Tunable, Flexible, and Efficient Optimization of Control Pulses for Practical Qubits,”Phys. Rev. Lett.120, 150401 (2018). doi:10.1103/PhysRevLett.120.150401

work page doi:10.1103/physrevlett.120.150401 2018
[52]

Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape,

N. Rach, M. M. M¨ uller, T. Calarco, and S. Montangero, “Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape,”Phys. Rev. A92, 062343 (2015). doi:10.1103/PhysRevA.92.062343

work page doi:10.1103/physreva.92.062343 2015
[53]

The roles of drift and control field constraints upon quantum control speed limits,

C. Arenz, B. Russell, D. Burgarth, and H. Rabitz, “The roles of drift and control field constraints upon quantum control speed limits,”New J. Phys.19, 103015 (2017). doi:10.1088/1367-2630/aa8242 19

work page doi:10.1088/1367-2630/aa8242 2017