- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Pontryagin maximum principle for Rydberg-blockaded state-to-state transfers: A semianalytic approach

Federico Astolfi*, Sven Jandura, and Guido Pupillo†

  • *Contact author: astolfi@unistra.fr
  • †Contact author: pupillo@unistra.fr
PDF

Phys. Rev. Research 8, 023162 – Published 14 May, 2026

DOI: https://doi.org/10.1103/2w2v-hpst

Abstract

We study time-optimal state-to-state control for two- and multiqubit operations motivated by neutral-atom quantum processors within the Rydberg blockade regime. Block diagonalization of the Hamiltonian simplifies the dynamics and enables the application of a semianalytic approach to the Pontryagin   maximum principle   (PMP) to derive optimal laser controls. We provide a general formalism for N qubits. For N=2 qubits, we classify normal and abnormal extremals, showcasing examples where abnormal solutions are either absent or suboptimal. For normal extremals, we establish a correspondence between the laser detuning from atomic transitions and the motion of a classical particle in a quartic potential, yielding a reduced, semianalytic formulation of the control problem. Combining PMP-based insights with numerical optimization, our approach bridges analytic and computational methods for high-fidelity, time-optimal control.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (67)

  1. F. Santambrogio, A Course in the Calculus of Variations: Optimization, Regularity, and Modeling (Springer, Cham, 2023).
  2. L. S. Pontryagin, The Mathematical Theory of Optimal Processes (Interscience Publishers, New York, 1962).
  3. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint (Springer, Berlin, 2004).
  4. H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples (Springer, New York, 2012).
  5. 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).
  6. L. Casalino and L. Mascolo, Pontryagin's principle for the optimization of escape trajectories from Earth–Moon L2, in New Trends and Challenges in Optimization Theory Applied to Space Engineering, edited by P. Cannarsa, A. Celletti, G. Fasano, L. Mazzini, M. Pontani, and E. Trélat, Springer Optimization and Its Applications (Springer, Cham, 2025), Vol. 221, pp. 9–22.
  7. S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, C. Kühn, I. Kuprov, B. Luy, S. G. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Training Schrödinger's cat: quantum optimal control, Eur. Phys. J. D 69, 279 (2015).
  8. M. H. Goerz, T. Calarco, and C. P. Koch, The quantum speed limit of optimal controlled phase gates for trapped neutral atoms, J. Phys. B: At. Mol. Opt. Phys. 44, 154011 (2011).
  9. M. M. Müller, D. M. Reich, M. Murphy, H. Yuan, J. Vala, K. B. Whaley, T. Calarco, and C. P. Koch, Optimizing entangling quantum gates for physical systems, Phys. Rev. A 84, 042315 (2011).
  10. M. H. Goerz, E. J. Halperin, J. M. Aytac, C. P. Koch, and K. B. Whaley, Robustness of high-fidelity Rydberg gates with single-site addressability, Phys. Rev. A 90, 032329 (2014).
  11. A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Generation and manipulation of Schrödinger cat states in Rydberg atom arrays, Science 365, 570 (2019).
  12. J. Cui, R. van Bijnen, T. Pohl, S. Montangero, and T. Calarco, Optimal control of Rydberg lattice gases, Quantum Sci. Technol. 2, 035006 (2017).
  13. A. Smith, B. E. Anderson, H. Sosa-Martinez, C. A. Riofrío, I. H. Deutsch, and P. S. Jessen, Quantum control in the Cs 6s1/2 ground manifold using radio-frequency and microwave magnetic fields, Phys. Rev. Lett. 111, 170502 (2013).
  14. B. E. Anderson, H. Sosa-Martinez, C. A. Riofrío, Ivan H. Deutsch, and Poul S. Jessen, Accurate and robust unitary transformations of a high-dimensional quantum system, Phys. Rev. Lett. 114, 240401 (2015).
  15. V. Nebendahl, H. Haffner, and C. F. Roos, Optimal control of entangling operations for trapped ion quantum computing, Phys. Rev. A 79, 012312 (2009).
  16. T. Choi, S. Debnath, T. A. Manning, C. Figgatt, Z.-X. Gong, L.-M. Duan, and C. Monroe, Optimal quantum control of multimode couplings between trapped ion qubits for scalable entanglement, Phys. Rev. Lett. 112, 190502 (2014).
  17. D. J. Egger and F. K. Wilhelm, Optimized controlled z gates for two superconducting qubits coupled through a resonator, Supercond. Sci. Technol. 27, 014001 (2014).
  18. J. Kelly, et al., Optimal quantum control using randomized benchmarking, Phys. Rev. Lett. 112, 240504 (2014).
  19. S.-Y. Huang and H.-S. Goan, Optimal control for fast and high-fidelity quantum gates in coupled superconducting flux qubits, Phys. Rev. A 90, 012318 (2014).
  20. M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, and S. Filipp, Leakage reduction in fast superconducting qubit gates via optimal control, npj Quantum Inf. 7, 14 (2021).
  21. D. D’ Alessandro, Introduction to Quantum Control and Dynamics, 2nd ed. (Chapman and Hall/CRC, New York, 2021).
  22. Q. Ansel, E. Dionis, F. Arrouas, B. Peaudecerf, S. Guérin, D. Guéry-Odelin, and D. Sugny, Introduction to theoretical and experimental aspects of quantum optimal control, J. Phys. B: At. Mol. Opt. Phys. 57, 133001 (2024).
  23. A. Garon, S. J. Glaser, and D. Sugny, Time-optimal control of SU(2) quantum operations, Phys. Rev. A 88, 043422 (2013).
  24. F. Albertini and D. D’ Alessandro, Minimum time optimal synthesis for two level quantum systems, J. Math. Phys. 56, 012106 (2015).
  25. O. Fresse-Colson, S. Guérin, Xi Chen, and D. Sugny, Application of the Pontryagin Maximum Principle to the robust time-optimal control of two-level quantum systems, Phys. Rev. A 112, 022618 (2025).
  26. R. Romano, Geometric analysis of minimum-time trajectories for a two-level quantum system, Phys. Rev. A 90, 062302 (2014).
  27. U. Boscain, M. Sigalotti, and D. Sugny, Introduction to the Pontryagin maximum principle for quantum optimal control, PRX Quantum 2, 030203 (2021).
  28. N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005).
  29. D. M. Reich, M. Ndong, and C. P. Koch, Monotonically convergent optimization in quantum control using Krotov's method, J. Chem. Phys. 136, 104103 (2012).
  30. J. Werschnik and E. K. U. Gross, Quantum optimal control theory, J. Phys. B: At. Mol. Opt. Phys. 40, R175 (2007).
  31. S. Jandura and G. Pupillo, Time-optimal two- and three-qubit gates for Rydberg atoms, Quantum, 6, 712 (2022).
  32. C. Fromonteil, R. Tricarico, F. Cesa, and H. Pichler, Hamilton-Jacobi-Bellman equations for Rydberg-blockade processes, Phys. Rev. Res. 6, 033333 (2024).
  33. M. Bergonzoni, S. Jandura, and G. Pupillo, iswap gate with polar molecules: Robustness criteria for entangling operations, Phys. Rev. A 112, 032621 (2025).
  34. M. Mohan, R. de Keijzer, and S. Kokkelmans, Robust control and optimal Rydberg states for neutral atom two-qubit gates, Phys. Rev. Res. 5, 033052 (2023).
  35. R. J. P. T. de Keijzer, L. Y. Visser, O. Tse, and S. J. J. M. F. Kokkelmans, Qubit fidelity distribution under stochastic Schrödinger equations driven by classical noise, Phys. Rev. Res. 7, 023063 (2025).
  36. R. J. P. T. de Keijzer, L. Y. Visser, O. Tse, and S. J. J. M. F. Kokkelmans, Fidelity-enhanced variational quantum optimal control, Phys. Rev. A 111, 052625 (2025).
  37. A. Delakouras, G. Doultsinos, and D. Petrosyan, Multi-qubit Rydberg gates between distant atoms, Quantum, 10, 1990 (2026).
  38. T. F. Gallagher, Rydberg Atoms (Cambridge Monographs on Atomic, Molecular and Chemical Physics) (Cambridge University Press, Cambridge, 1994).
  39. M. Morgado and S. Whitlock, Quantum simulation and computing with Rydberg-interacting qubits, AVS Quantum Sci. 3, 023501 (2021).
  40. M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  41. M. Saffman, Quantum computing with atomic qubits and Rydberg interactions: progress and challenges, J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016).
  42. H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletić, H. Pichler, and M. D. Lukin, Parallel implementation of high-fidelity multiqubit gates with neutral atoms, Phys. Rev. Lett. 123, 170503 (2019).
  43. S. Ma, G. Liu, P. Peng, S. Jandura, G. Pupillo, et al., High-fidelity gates and mid-circuit erasure conversion in an atomic qubit, Nature (London) 622, 279 (2023).
  44. T. M. Graham, M. Kwon, B. Grinkemeyer, Z. Marra, X. Jiang, M. T. Lichtman, Y. Sun, M. Ebert, and M. Saffman, Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array, Phys. Rev. Lett. 123, 230501 (2019).
  45. A. Pagano, S. Weber, D. Jaschke, T. Pfau, F. Meinert, S. Montangero, and H. P. Büchler, Error-budgeting for a controlled-phase gate with Strontium-88 Rydberg atoms, Phys. Rev. Res. 4, 033019 (2022).
  46. S. Jandura, J. D. Thompson, and G. Pupillo, Optimizing Rydberg gates for logical qubit performance, PRX Quantum 4, 020336 (2023).
  47. S. J. Evered, D. Bluvstein, M. Kalinowski, et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature (London) 622, 268 (2023).
  48. A. G. Radnaev, W. C. Chung, D. C. Cole, D. Mason, T. G. Ballance, et al., Universal neutral-atom quantum computer with individual optical addressing and nondestructive readout, PRX Quantum 6, 030334 (2025).
  49. R. Tao, O. Lib, F. Gyger, H. Timme, M. Ammenwerth, I. Bloch, and J. Zeiher, Universal gates for a metastable qubit in Strontium-88, arXiv:2506.10714.
  50. J. A. Muniz, M. Stone, D. T. Stack, M. Jaffe, J. M. Kindem, et al., High-fidelity universal gates in the Yb171 ground-state nuclear-spin qubit, PRX Quantum 6, 020334 (2025).
  51. R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, Benchmarking and fidelity response theory of high-fidelity Rydberg entangling gates, PRX Quantum, 6, 010331 (2025).
  52. G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin, Probing topological spin liquids on a programmable quantum simulator, Science 374, 1242 (2021).
  53. A. W. Glätzle, M. Dalmonte, R. Nath, I. Rousochatzakis, R. Moessner, and P. Zoller, Quantum spin-ice and dimer models with Rydberg atoms, Phys. Rev. X 4, 041037 (2014).
  54. A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  55. T. Lahaye, C. Menotti, L. Santos, and M. Lewenstein, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72, 126401 (2009).
  56. D. González-Cuadra, M. Hamdan, T. V. Zache, et al., Observation of string breaking on a (2+1)d Rydberg quantum simulator, Nature (London) 642, 321 (2025).
  57. D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner, V. Vuletić, and M. D. Lukin, Controlling quantum many-body dynamics in driven Rydberg atom arrays, Science 371, 1355 (2021).
  58. H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macrì, T. Lahaye, and A. Browaeys, Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models, Nature (London) 534, 667 (2016).
  59. A. Cao, W. J. Eckner, T. Lukin, Y. Yelin, et al., Multi-qubit gates and Schrödinger cat states in an optical clock, Nature (London) 634, 315 (2024).
  60. S. Jandura, Optimized quantum gates for neutral atom quantum computers, Ph.D. thesis, 2024, https://theses.hal.science/tel-04995702.
  61. F. Cesa and H. Pichler, Universal quantum computation in globally driven Rydberg atom arrays, Phys. Rev. Lett. 131, 170601 (2023).
  62. Recall that Tq*(t)M≃RdimM.
  63. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed. (Academic Press, Amsterdam, 2003).
  64. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Cambridge University Press, Cambridge, 2010).
  65. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).
  66. See Ref. [31] for other symmetries relevant to the realization of general C-phase gates, including the cz gate considered in the state-to-state transfer of Case (ii).
  67. SciML. Optim.jl: Optimization algorithms (2025), https://docs.sciml.ai/Optimization/stable/optimization_packages/optim/.

Outline

Information

Phys. Rev. Research 8, 023162– Published 14 May, 2026
  • Received 16 December 2025
  • Accepted 13 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation