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arxiv: 2509.18737 · v2 · submitted 2025-09-23 · 🪐 quant-ph · cond-mat.mes-hall· math-ph· math.MP· physics.atom-ph

Overcoming limitations on gate fidelity in noisy static exchange-coupled surface qubits

Hoang-Anh Le , Saba Taherpour , Denis Jankovi\'c , Christoph Wolf This is my paper

Pith reviewed 2026-05-18 15:07 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallmath-phmath.MPphysics.atom-ph
keywords surface qubitsexchange couplingquantum optimal controlKrotov methodgate fidelityopen quantum systemsatomic spin resonanceall-electric ESR
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The pith

Quantum optimal control pulses can reach gate fidelities above 0.9 in static exchange-coupled surface qubits by adapting to noise, outperforming standard Rabi driving.

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

Surface qubits formed by individual atom spins controlled via all-electric resonance suffer from fixed exchange couplings between neighbors, short coherence lifetimes, and imperfect initial polarization. These constraints limit conventional driving methods to low gate fidelities. The paper uses open-quantum-system simulations to test whether the Krotov optimal-control algorithm can generate tailored pulses that compensate for the specific noise and coupling. Results show that fidelities of 0.9 or higher are reachable once pulse parameters and experimental conditions are chosen appropriately, and that these pulses beat ordinary Rabi drives. The authors close by suggesting concrete changes to the original experimental geometry to realize the higher fidelities.

Core claim

In open-system simulations of exchange-coupled surface qubits, the Krotov quantum optimal control method produces gate operations with fidelity F ≳ 0.9 when noise spectra, lifetimes, and initial-state polarization are accounted for; the resulting pulses outperform conventional Rabi driving and point to redesigned experimental setups that maximize fidelity.

What carries the argument

Krotov iterative optimal-control algorithm applied to the Lindblad master equation of the exchange-coupled spin system.

If this is right

  • Gate fidelities of 0.9 or higher become attainable once control pulses are optimized for the measured noise and coupling strengths.
  • Krotov pulses systematically outperform fixed-frequency Rabi driving for the same platform parameters.
  • Modest changes to the original experimental geometry and pulse shaping can push achievable fidelity higher.

Where Pith is reading between the lines

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

  • The same optimal-control strategy could be applied to other atom-on-surface or fixed-coupling qubit arrays where conventional driving is constrained.
  • Multi-qubit entangling gates in this platform would likely require similar pulse optimization rather than simple extensions of single-qubit Rabi drives.
  • If the simulated fidelity gains hold in experiment, surface-atom arrays could become a viable route to small-scale quantum processors without needing tunable couplers.

Load-bearing premise

The open quantum system model used in the simulations accurately captures the dominant noise sources, lifetime limits, and initial-state polarization present in the actual experimental surface-qubit platform.

What would settle it

Implement the Krotov-derived pulses on the same surface-atom hardware used in the original experiments and measure the resulting two-qubit gate fidelity; a value well below 0.9 while the measured noise and lifetime parameters remain close to the simulated values would falsify the claim.

Figures

Figures reproduced from arXiv: 2509.18737 by Christoph Wolf, Denis Jankovi\'c, Hoang-Anh Le, Saba Taherpour.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: summarizes how energy relaxation and pure dephasing reshapes the spectrum of the Krotov￾optimized control. Throughout, we quantify the width w by the full width at half maximum (FWHM) of each spectral line and the height h by the corresponding peak amplitude. As in the closed-system case discussed above, the optimal spectrum indicates that Krotov’s method ex￾ploits the full set of available control degrees… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

Recent experiments demonstrated that the spin state of individual atoms on surfaces can be quantum-coherently controlled through all-electric electron spin resonance. By constructing interacting arrays of atoms this results in an atomic-scale qubit platform. However, the static exchange coupling between qubits, limited lifetime and polarization of the initial state, impose significant limits on high-fidelity quantum control. We address this issue using open quantum systems simulation and quantum optimal control theory. We demonstrate the conditions under which high-fidelity operations ($\mathcal{F} \gtrsim 0.9$) are feasible in this qubit platform, and show how the Krotov method of quantum optimal control theory adapts to specific noise sources to outperform the conventional Rabi drivings. Finally, we re-examine the experimental setup used in the initial demonstration of this qubit platform and propose optimized experimental designs to maximize gate fidelity in this platform.

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

1 major / 2 minor

Summary. The paper uses open-quantum-system master-equation simulations of surface-atom qubits with static exchange coupling, finite lifetime, and partial initial-state polarization. It applies the Krotov optimal-control algorithm to design driving pulses that achieve gate fidelities F ≳ 0.9 and shows these pulses outperform conventional Rabi driving; the work concludes by proposing concrete modifications to the experimental geometry and pulse parameters of the original surface-qubit demonstration.

Significance. If the noise model is representative, the results supply a practical route to usable two-qubit gates in an atomic-scale platform whose main decoherence channels are already identified. The explicit comparison of Krotov versus Rabi under the same Lindblad operators, together with the re-optimization of the existing experimental layout, constitutes a concrete, falsifiable prediction that experimental groups can test directly.

major comments (1)
  1. [Simulation section (model definition and parameter table)] The central fidelity threshold F ≳ 0.9 and the claimed superiority of Krotov control rest entirely on the open-system model (Lindblad operators, decoherence rates, and initial polarization). No section compares these rates or the resulting T1, T2 values to the experimental numbers reported in the referenced surface-qubit demonstration; without this anchoring the predicted feasibility window remains unverified.
minor comments (2)
  1. [Figures 3 and 4] Figure captions should state the precise pulse bandwidth and the number of Krotov iterations used to reach the reported fidelities.
  2. [Abstract] The abstract states that the Krotov method 'adapts to specific noise sources'; a one-sentence summary of which noise operators are actively compensated would help readers immediately see the mechanism.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The suggestion to explicitly anchor the noise model to experimental data is well taken and will improve the clarity of the work. We address the major comment below.

read point-by-point responses

  1. Referee: [Simulation section (model definition and parameter table)] The central fidelity threshold F ≳ 0.9 and the claimed superiority of Krotov control rest entirely on the open-system model (Lindblad operators, decoherence rates, and initial polarization). No section compares these rates or the resulting T1, T2 values to the experimental numbers reported in the referenced surface-qubit demonstration; without this anchoring the predicted feasibility window remains unverified.

    Authors: We agree that an explicit comparison is necessary to verify the model. The Lindblad rates and initial polarization in our simulations are taken directly from the experimental parameters of the referenced surface-qubit demonstration (finite lifetime and partial polarization). However, the manuscript does not contain a dedicated comparison of the resulting T1 and T2 times. In the revised version we will add a new paragraph and table in the Simulation section that (i) lists the experimental T1/T2 values from the reference, (ii) shows the mapping to our chosen decoherence rates, and (iii) discusses any approximations involved. This addition will make the feasibility window directly falsifiable against the cited experiment. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical predictions from explicit open-system model

full rationale

The paper applies standard Lindblad master equations and Krotov optimal control numerics to a parameterized model of static exchange coupling, lifetime, and initial polarization. The reported fidelities (F ≳ 0.9) and claimed advantage over Rabi driving are direct outputs of these simulations under the stated assumptions; they do not reduce by construction to any fitted parameter or self-citation that encodes the target result. The derivation chain is therefore self-contained and externally falsifiable against future experiments on the physical platform.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard quantum-optics master equations and optimal-control theory; no new free parameters, axioms, or invented entities are introduced beyond the usual modeling assumptions for open quantum systems.

axioms (1)
  • domain assumption The dynamics of the surface qubits are well-described by a Lindblad master equation with Markovian noise.
    Invoked when the authors model the platform as an open quantum system.

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Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    mpz4XXru8WOM6CHfr91sbk8G5fU=

    In addition to the fidelity between Bell state and the final state (obtained via Krotov method), we use concurrence Cof the final state as an additional metric. Concurrence quantifies entanglement of two qubits: concurrence of fully separated states is zero and concurrence of maximally entangled states such as Bell states is one. Figures 4(a) and (b) pres...

    work page

  2. [2]

    Y. Wang, Y. Chen, H. T. Bui, C. Wolf, M. Haze, C. Mier, J. Kim, D.-J. Choi, C. P. Lutz, Y. Bae, S. hyon Phark, and A. J. Heinrich, An atomic-scale multi-qubit platform, Science382, 87 (2023)

    work page 2023

  3. [3]

    C. Wolf, A. J. Heinrich, and S.-h. Phark, On-surface atomic scale qubit platform, ACS nano18, 28469 (2024)

    work page 2024

  4. [4]

    Y. Chen, Y. Bae, and A. J. Heinrich, Harnessing the quantum behavior of spins on surfaces, Advanced Mate- rials35, 2107534 (2023)

    work page 2023

  5. [5]

    Baumann, W

    S. Baumann, W. Paul, T. Choi, C. P. Lutz, A. Ardavan, and A. J. Heinrich, Electron paramagnetic resonance of individual atoms on a surface, Science350, 417 (2015)

    work page 2015

  6. [6]

    T. S. Seifert, S. Kovarik, D. M. Juraschek, N. A. Spaldin, P. Gambardella, and S. Stepanow, Longitudinal and transverse electron paramagnetic resonance in a scan- ning tunneling microscope, Science advances6, eabc5511 (2020)

    work page 2020

  7. [7]

    K. Yang, W. Paul, S.-H. Phark, P. Willke, Y. Bae, T. Choi, T. Esat, A. Ardavan, A. J. Heinrich, and C. P. Lutz, Coherent spin manipulation of individual atoms on a surface, Science366, 509 (2019)

    work page 2019

  8. [8]

    Willke, T

    P. Willke, T. Bilgeri, X. Zhang, Y. Wang, C. Wolf, H. Aubin, A. Heinrich, and T. Choi, Coherent spin con- trol of single molecules on a surface, ACS nano15, 17959 (2021)

    work page 2021

  9. [9]

    Y. Wang, M. Haze, H. T. Bui, W.-h. Soe, H. Aubin, A. Ardavan, A. J. Heinrich, and S.-h. Phark, Universal quantum control of an atomic spin qubit on a surface, npj Quantum Information9, 48 (2023)

    work page 2023

  10. [10]

    Phark, H

    S.-h. Phark, H. T. Bui, A. Ferr´ on, J. Fern´ andez-Rossier, J. Reina-G´ alvez, C. Wolf, Y. Wang, K. Yang, A. J. Hein- rich, and C. P. Lutz, Electric-field-driven spin resonance by on-surface exchange coupling to a single-atom magnet, Advanced Science10, 2302033 (2023)

    work page 2023

  11. [11]

    Phark, Y

    S.-h. Phark, Y. Chen, H. T. Bui, Y. Wang, M. Haze, J. Kim, Y. Bae, A. J. Heinrich, and C. Wolf, Double- 12 resonance spectroscopy of coupled electron spins on a surface, ACS Nano17, 14144 (2023)

    work page 2023

  12. [12]

    H. T. Bui, C. Wolf, Y. Wang, M. Haze, A. Ardavan, A. J. Heinrich, and S.-h. Phark, All-electrical driving and probing of dressed states in a single spin, ACS Nano18, 12187 (2024)

    work page 2024

  13. [13]

    Choi, S.-h

    D.-J. Choi, S.-h. Phark, A. J. Heinrich, and N. Lorente, Electron spin resonance with scanning tunneling mi- croscopy: a tool for an on-surface quantum platform of identical qubits, arXiv preprint arXiv:2505.10079 10.48550/arXiv.2505.10079 (2025)

    work page doi:10.48550/arxiv.2505.10079 2025

  14. [14]

    Rigetti and M

    C. Rigetti and M. Devoret, Fully microwave-tunable uni- versal gates in superconducting qubits with linear cou- plings and fixed transition frequencies, Phys. Rev. B81, 134507 (2010)

    work page 2010

  15. [15]

    Warren, U

    A. Warren, U. G¨ ung¨ ord¨ u, J. P. Kestner, E. Barnes, and S. E. Economou, Robust photon-mediated entangling gates between quantum dot spin qubits, Phys. Rev. B 104, 115308 (2021)

    work page 2021

  16. [16]

    Heinz and G

    I. Heinz and G. Burkard, Crosstalk analysis for single- qubit and two-qubit gates in spin qubit arrays, Phys. Rev. B104, 045420 (2021)

    work page 2021

  17. [17]

    L. M. K. Vandersypen and I. L. Chuang, Nmr techniques for quantum control and computation, Rev. Mod. Phys. 76, 1037 (2005)

    work page 2005

  18. [18]

    Willke, W

    P. Willke, W. Paul, F. D. Natterer, K. Yang, Y. Bae, T. Choi, J. Fern´ andez-Rossier, A. J. Heinrich, and C. P. Lutz, Probing quantum coherence in single-atom electron spin resonance, Science Advances4, eaaq1543 (2018)

    work page 2018

  19. [20]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2007)

    work page 2007

  20. [21]

    Morvan, L

    A. Morvan, L. Chen, J. M. Larson, D. I. Santiago, and I. Siddiqi, Optimizing frequency allocation for fixed- frequency superconducting quantum processors, Phys. Rev. Res.4, 023079 (2022)

    work page 2022

  21. [22]

    Zhao, Mitigation of quantum crosstalk in cross- resonance-based qubit architectures, Phys

    P. Zhao, Mitigation of quantum crosstalk in cross- resonance-based qubit architectures, Phys. Rev. Appl. 20, 054033 (2023)

    work page 2023

  22. [23]

    Heinz and G

    I. Heinz and G. Burkard, High-fidelity cnot gate for spin qubits with asymmetric driving using virtual gates, Phys. Rev. B105, L121402 (2022)

    work page 2022

  23. [24]

    D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-free subspaces for quantum computation, Phys. Rev. Lett.81, 2594 (1998)

    work page 1998

  24. [25]

    Y. Bae, K. Yang, P. Willke, T. Choi, A. J. Heinrich, and C. P. Lutz, Enhanced quantum coherence in exchange coupled spins via singlet-triplet transitions, Science Ad- vances4, eaau4159 (2018)

    work page 2018

  25. [26]

    Viola, E

    L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R. Laflamme, and D. G. Cory, Experimental realization of noiseless subsystems for quantum information process- ing, Science293, 2059 (2001)

    work page 2059

  26. [27]

    Viola, E

    L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Phys. Rev. Lett.82, 2417 (1999)

    work page 1999

  27. [28]

    Clausen, G

    J. Clausen, G. Bensky, and G. Kurizki, Bath-optimized minimal-energy protection of quantum operations from decoherence, Phys. Rev. Lett.104, 040401 (2010)

    work page 2010

  28. [29]

    C. Brif, R. Chakrabarti, and H. Rabitz, Control of quan- tum phenomena: past, present and future, New Journal of Physics12, 075008 (2010)

    work page 2010

  29. [30]

    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. K¨ ockenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbr¨ uggen,et al., Training schr¨ odinger’s cat: Quantum optimal control: Strategic report on current status, visions and goals for research in europe, The European Physical Journal D69, 1 (2015)

    work page 2015

  30. [31]

    J. P. Palao and R. Kosloff, Quantum computing by an optimal control algorithm for unitary transformations, Phys. Rev. Lett.89, 188301 (2002)

    work page 2002

  31. [32]

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

    work page 2003

  32. [33]

    Khaneja, T

    N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen, and S. J. Glaser, Optimal control of coupled spin dy- namics: design of nmr pulse sequences by gradient as- cent algorithms, Journal of Magnetic Resonance172, 296 (2005)

    work page 2005

  33. [34]

    T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja, and S. J. Glaser, Application of optimal control theory to the design of broadband excitation pulses for high-resolution nmr, Journal of Magnetic Resonance163, 8 (2003)

    work page 2003

  34. [35]

    Castro, A

    A. Castro, A. Garc´ ıa Carrizo, S. Roca, D. Zueco, and F. Luis, Optimal control of molecular spin qudits, Phys. Rev. Appl.17, 064028 (2022)

    work page 2022

  35. [36]

    V. F. Krotov and I. Feldman, An iterative method for solving optimal control problems, Eng. Cybern21, 123 (1983)

    work page 1983

  36. [37]

    Soml´ oi, V

    J. Soml´ oi, V. A. Kazakov, and D. J. Tannor, Controlled dissociation of i2 via optical transitions between the x and b electronic states, Chemical Physics172, 85 (1993)

    work page 1993

  37. [38]

    D. M. Reich, M. Ndong, and C. P. Koch, Monotoni- cally convergent optimization in quantum control using krotov’s method, The Journal of chemical physics136, doi:10.1063/1.3691827 (2012)

    work page doi:10.1063/1.3691827 2012

  38. [39]

    X.-L. He, J. Q. You, Y.-x. Liu, L. F. Wei, and F. Nori, Switchable coupling between charge and flux qubits, Phys. Rev. B76, 024517 (2007)

    work page 2007

  39. [40]

    Burkard, T

    G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Rev. Mod. Phys.95, 025003 (2023)

    work page 2023

  40. [41]

    M. Russ, D. M. Zajac, A. J. Sigillito, F. Borjans, J. M. Taylor, J. R. Petta, and G. Burkard, High-fidelity quan- tum gates in si/sige double quantum dots, Phys. Rev. B 97, 085421 (2018)

    work page 2018

  41. [42]

    Nosrati, A

    F. Nosrati, A. Castellini, G. Compagno, and R. Lo Franco, Robust entanglement preparation against noise by controlling spatial indistinguishability, npj Quantum Information6, 39 (2020)

    work page 2020

  42. [43]

    H. Yu, Z. Wang, and J.-H. Wu, Entanglement prepa- ration and nonreciprocal excitation evolution in giant atoms by controllable dissipation and coupling, Phys. Rev. A104, 013720 (2021)

    work page 2021

  43. [44]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge University Press, 2010)

    work page 2010

  44. [45]

    Verstraete and H

    F. Verstraete and H. Verschelde, Fidelity of mixed states of two qubits, Phys. Rev. A66, 022307 (2002)

    work page 2002

  45. [46]

    M. H. Goerz, D. M. Reich, and C. P. Koch, Optimal control theory for a unitary operation under dissipative evolution, New Journal of Physics16, 055012 (2014)

    work page 2014

  46. [47]

    Jankovi´ c, J.-G

    D. Jankovi´ c, J.-G. Hartmann, M. Ruben, and P.-A. Hervieux, Noisy qudit vs multiple qubits: conditions on gate efficiency for enhancing fidelity, npj Quantum Infor- mation10, 10.1038/s41534-024-00829-6 (2024). 13

    work page doi:10.1038/s41534-024-00829-6 2024

  47. [48]

    Hartmann, D

    J.-G. Hartmann, D. Jankovi´ c, R. Pasquier, M. Ruben, and P.-A. Hervieux, Nonlinearity of the fidelity in open qudit systems: Gate and noise dependence in high- dimensional quantum computing, Quantum9, 1690 (2025)

    work page 2025

  48. [49]

    I. I. Sobel’Man, L. A. Vainshtein, and E. A. Yukov,Exci- tation of atoms and broadening of spectral lines, Vol. 15 (Springer Science & Business Media, 2012)

    work page 2012

  49. [50]

    hyon Phark, H

    S. hyon Phark, H. T. Bui, W. hyo Seo, Y. Liu, V. Sheina, C. Lee, C. Wolf, A. J. Heinrich, R. Rob- les, and N. Lorente, Spin-state engineering of single ti- tanium adsorbates on ultrathin magnesium oxide (2025), arXiv:2507.23299 [cond-mat.mes-hall]

    work page arXiv 2025

  50. [51]

    W. Paul, K. Yang, S. Baumann, N. Romming, T. Choi, C. P. Lutz, and A. J. Heinrich, Control of the millisec- ond spin lifetime of an electrically probed atom, Nature Physics13, 403 (2017)

    work page 2017

  51. [52]

    Broekhoven, C

    R. Broekhoven, C. Lee, S.-h. Phark, S. Otte, and C. Wolf, Protocol for certifying entanglement in surface spin sys- tems using a scanning tunneling microscope, npj Quan- tum Information10, 92 (2024)

    work page 2024

  52. [53]

    L. H. Pedersen, N. M. Møller, and K. Mølmer, Fidelity of quantum operations, Physics Letters A367, 47 (2007)

    work page 2007

  53. [54]

    Johansson, P

    J. Johansson, P. Nation, and F. Nori, Qutip: An open- source python framework for the dynamics of open quan- tum systems, Computer Physics Communications183, 1760 (2012)

    work page 2012

  54. [55]

    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, 80 (2019)

    work page 2019

  55. [56]

    M. A. Nielsen, A simple formula for the average gate fi- delity of a quantum dynamical operation, Physics Letters A303, 249 (2002)

    work page 2002

  56. [57]

    ImgedVgviPXDFI3KWMd9V5IhmEc=

    W. K. Wootters, Entanglement of formation of an ar- bitrary state of two qubits, Phys. Rev. Lett.80, 2245 (1998). 14 Supplemental Material Overcoming limitations on gate fidelity in noisy static exchange-coupled surface qubits For most of the simulations we used a drift Hamiltonian withω 1 = 20πGHz, ω 2 = 14πGHz, J= 5 GHz. This Hamiltonian is partially op...

    work page 1998