- 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

Harnessing subspace controllability: Dynamical generation of Dicke states in Heisenberg-coupled qubit arrays with a single local control

Vladimir M. Stojanović1, Tommaso Calarco2,3,4, and Andrea Muratori4

PDF

Phys. Rev. Research 8, 033095 – Published 23 July, 2026

DOI: https://doi.org/10.1103/t2zk-4nxh

Abstract

We explore the feasibility of realizing Dicke states in qubit arrays with always-on isotropic Heisenberg coupling between adjacent qubits, assuming a single Zeeman-type control acting in the z direction on an actuator qubit. The Lie-algebraic criteria of controllability imply that such an array is not completely controllable, but satisfies the conditions for subspace controllability on any subspace with a fixed number of excitations. Therefore, a qubit array described by the model under consideration is state-to-state controllable for an arbitrary choice of initial and final states that have the same Hamming weight. This limited controllability is exploited here for the time-efficient dynamical generation of an a-excitation Dicke state |DaN〉 (a=1,2,...,N−1) in a linear array with N qubits starting from a generic Hamming-weight-a product state. To dynamically generate the desired Dicke states—including W states |WN〉 as their special (a=1) case—in the shortest possible time with a single local Z control, we employ an optimal-control scheme based on the dressed chopped random basis (dCRAB) algorithm. We optimize the target-state fidelity over the expansion coefficients of smoothly varying control fields in a truncated random Fourier basis; this is done by combining Nelder-Mead-type local optimizations with the multistart-based clustering algorithm that facilitates searches for global extrema. In this manner, we obtain the optimal-control fields for Dicke-state preparation in arrays with up to nine qubits. Based on our numerical results, we find that the shortest possible state-preparation times scale as O(N2.08) for W states and O(N1.78) for a=2 Dicke states. Finally, we demonstrate the robustness of our dCRAB-based state-engineering scheme against various types of imperfections of relevance for its anticipated experimental implementation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (90)

  1. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  2. M. Bergmann and O. Gühne, J. Entanglement criteria for Dicke states, Phys. A: Math. Theor. 46, 385304 (2013).
  3. A. Neven, J. Martin, and T. Bastin, Entanglement robustness against particle loss in multiqubit systems, Phys. Rev. A 98, 062335 (2018).
  4. W. Zhang, Z. Han, F. Shi, and X. Zhang, Constructions of multipartite entanglement resistant to particle loss, Phys. Rev. A 112, 022408 (2025).
  5. D. A. Lidar and K. B. Whaley, Decoherence-free subspaces and subsystems, in Irreversible Quantum Dynamics, edited by F. Benatti and R. Floreanini (Springer, Berlin, 2003), pp. 83–120.
  6. V. M. Stojanović and J. K. Nauth, Dicke-state preparation through global transverse control of Ising-coupled qubits, Phys. Rev. A 108, 012608 (2023).
  7. S. K. Özdemir, J. Shimamura, and N. Imoto, A necessary and sufficient condition to play games in quantum mechanical settings, New J. Phys. 9, 43 (2007).
  8. R. Prevedel, G. Cronenberg, M. S. Tame, M. Paternostro, P. Walther, M. S. Kim, and A. Zeilinger, Experimental realization of Dicke states of up to six qubits for multiparty quantum networking, Phys. Rev. Lett. 103, 020503 (2009).
  9. G. Toth, Multipartite entanglement and high-precision metrology, Phys. Rev. A 85, 022322 (2012).
  10. Z. H. Saleem, M. Perlin, A. Shaji, and S. K. Gray, Achieving the Heisenberg limit with Dicke states in noisy quantum metrology, Phys. Rev. A 109, 052615 (2024).
  11. See, e.g., Y. Ouyang, Permutation-invariant quantum codes, Phys. Rev. A 90, 062317 (2014).
  12. J. Golden, A. Bärtschi, D. O’ Malley, and S. Eidenbenz, Threshold-based quantum optimization, in Proceedings of the IEEE International Conference on Quantum Computing and Engineering (IEEE, Broomfield, CO, 2021), p. 137.
  13. W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Toth, and H. Weinfurter, Experimental entanglement of a six-photon symmetric Dicke state, Phys. Rev. Lett. 103, 020504 (2009).
  14. D. B. Hume, C. W. Chou, T. Rosenband, and D. J. Wineland, Preparation of Dicke states in an ion chain, Phys. Rev. A 80, 052302 (2009).
  15. L. Lamata, C. E. López, B. P. Lanyon, T. Bastin, J. C. Retamal, and E. Solano, Deterministic generation of arbitrary symmetric states and entanglement classes, Phys. Rev. A 87, 032325 (2013).
  16. X.-Q. Shao, L. Chen, S. Zhang, Y.-F. Zhao, and K.-H. Yeon, Deterministic generation of arbitrary multi-atom symmetric Dicke states by a combination of quantum Zeno dynamics and adiabatic passage, Europhys. Lett. 90, 50003 (2010).
  17. V. Srivastava, S. Jandura, G. K. Brennen, and G. Pupillo, Entanglement-enhanced quantum sensing via optimal global control with neutral atoms in a cavity, Phys. Rev. Lett. 136, 060806 (2026).
  18. C.-Y. Zhang and J. Jing, Efficient nonclassical state preparation via generalized parity measurement, Phys. Rev. A 113, 022420 (2026).
  19. C. Wu, C. Guo, Y. Wang, G. Wang, X.-L. Feng, and J.-L. Chen, Generation of Dicke states in the ultrastrong-coupling regime of circuit QED systems, Phys. Rev. A 95, 013845 (2017).
  20. T. Kobayashi, R. Ikuta1, Ş. Kaya Özdemir, M. Tame, T. Yamamoto, M. Koashi, and N. Imoto, Universal gates for transforming multipartite entangled Dicke states, New J. Phys. 16, 023005 (2014).
  21. F. Albarran-Arriagada, G. Romero, and J. C. Retamal, Equally entangled multiqubit states, Phys. Scr. 101, 115102 (2026).
  22. V. Jurdjević and H. J. Sussmann, Control systems on Lie groups, J. Differ. Equations 12, 313 (1972).
  23. G. M. Huang, T. J. Tarn, and J. W. Clark, On the controllability of quantum-mechanical systems, J. Math. Phys. 24, 2608 (1983).
  24. V. Ramakrishna and H. Rabitz, Relation between quantum computing and quantum controllability, Phys. Rev. A 54, 1715 (1996).
  25. D. D’Alessandro, Introduction to Quantum Control and Dynamics (Taylor & Francis, Boca Raton, 2008).
  26. M. M. Müller, R. S. Said, F. Jelezko, T. Calarco, and S. Montangero, One decade of quantum optimal control in the chopped random basis, Rep. Prog. Phys. 85, 076001 (2022).
  27. Q. Ansel, E. Dionis, F. Arrouas, B. Peaudecerf, S. Guerin, D. Guery-Odelin, and D. Sugny, Introduction to theoretical and experimental aspects of quantum optimal control, J. Phys. B: At. Mol. Opt. Phys. 57, 133001 (2024).
  28. For a recent review, see C. W. Duncan, P. M. Poggi, M. Bukov, N. T. Zinner, and S. Campbell, Taming quantum systems: A tutorial for using shortcuts-to-adiabaticity, quantum optimal control, and reinforcement learning, PRX Quantum 6, 040201 (2025).
  29. P. Cappellaro, C. Ramanathan, and D. G. Cory, Dynamics and control of a quasi-1D spin system, Phys. Rev. Lett. 99, 250506 (2007).
  30. L. Stefanescu, L. Edwards-Pratt, J. O’Connor, E. Tsegaye, N. H. Le, and F. Mintert, Robust implicit quantum control of interacting spin chains, Phys. Rev. A 112, 012609 (2025).
  31. A. L. P. de Lima, L. S. Baker, A. Zlotnik, A. K. Harter, M. J. Martin, and J.-S. Li, Robust quantum-state generation in symmetric spin networks, Phys. Rev. A 113, 052222 (2026).
  32. S. C. Benjamin and S. Bose, Quantum computing with an always-on Heisenberg interaction, Phys. Rev. Lett. 90, 247901 (2003).
  33. R. Heule, C. Bruder, D. Burgarth, and V. M. Stojanović, Local quantum control of Heisenberg spin chains, Phys. Rev. A 82, 052333 (2010).
  34. See, e.g., Z. Chen, W. Liu, Y. Ma, W. Sun, R. Wang, H. Wang, H. Xu, G. Xue, H. Yan, Z. Yang, et al., Efficient implementation of arbitrary two-qubit gates using unified control, Nat. Phys. 21, 1489 (2025), and references therein.
  35. Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, et al., Qubit architecture with high coherence and fast tunable coupling, Phys. Rev. Lett. 113, 220502 (2014).
  36. D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, and K. B. Whaley, Universal quantum computation with the exchange interaction, Nature (London) 408, 339 (2000).
  37. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
  38. J. Levy, Universal quantum computation with spin-1/2 pairs and Heisenberg exchange, Phys. Rev. Lett. 89, 147902 (2002).
  39. D. Bacon, J. Kempe, D. A. Lidar, and K. B. Whaley, Universal fault-tolerant quantum computation on decoherence-free subspaces, Phys. Rev. Lett. 85, 1758 (2000).
  40. T. Tanamoto, D. Becker, V. M. Stojanović, and C. Bruder, Preserving universal resources for one-way quantum computing, Phys. Rev. A 86, 032327 (2012).
  41. See, e.g., T. Tanamoto, V. M. Stojanović, C. Bruder, and D. Becker, Strategy for implementing stabilizer-based codes on solid-state qubits, Phys. Rev. A 87, 052305 (2013).
  42. S. G. Schirmer, I. C. H. Pullen, and P. J. Pemberton-Ross, Global controllability with a single local actuator, Phys. Rev. A 78, 062339 (2008).
  43. X. Wang, D. Burgarth, and S. Schirmer, Subspace controllability of spin-1/2 chains with symmetries, Phys. Rev. A 94, 052319 (2016).
  44. R. Heule, C. Bruder, D. Burgarth, and V. M. Stojanović, Controlling qubit arrays with anisotropic XXZ Heisenberg interaction by acting on a single qubit, Eur. Phys. J. D 63, 41 (2011).
  45. V. M. Stojanović, Feasibility of single-shot realizations of conditional three-qubit gates in exchange-coupled qubit arrays with local control, Phys. Rev. A 99, 012345 (2019).
  46. F. Albertini and D. D’Alessandro, Quantum subspace controllability implying full controllability, Linear Algebra Appl. 705, 207 (2025).
  47. N. Rach, M. M. Müller, T. Calarco, and S. Montangero, Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape, Phys. Rev. A 92, 062343 (2015).
  48. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1999).
  49. A. Törn and A. Žilinskas, Global Optimization, Lecture Notes in Computer Science, Vol. 350 (Springer, Berlin, 1989).
  50. T. Haase, G. Alber, and V. M. Stojanović, Dynamical generation of chiral W and Greenberger-Horne-Zeilinger states in laser-controlled Rydberg-atom trimers, Phys. Rev. Res. 4, 033087 (2022).
  51. V. M. Stojanović, Bare-excitation ground state of a spinless-fermion-boson model and W-state engineering in an array of superconducting qubits and resonators, Phys. Rev. Lett. 124, 190504 (2020).
  52. V. M. Stojanović, Scalable W-type entanglement resource in neutral-atom arrays with Rydberg-dressed resonant dipole-dipole interaction, Phys. Rev. A 103, 022410 (2021).
  53. J. Zheng, J. Peng, P. Tang, F. Li, and N. Tan, Unified generation and fast emission of arbitrary single-photon multimode W states, Phys. Rev. A 105, 062408 (2022).
  54. G. Q. Zhang, W. Feng, W. Xiong, Q. P. Su, and C. P. Yang, Generation of long-lived W states via reservoir engineering in dissipatively coupled systems, Phys. Rev. A 107, 012410 (2023).
  55. V. Srivastava, S. Jandura, G. K. Brennen, and G. Pupillo, Cavity polariton blockade for non-local entangling gates with trapped atoms, Quantum Sci. Technol. 10, 045038 (2025).
  56. V. M. Stojanović and J. K. Nauth, Interconversion of W and Greenberger-Horne-Zeilinger states for Ising-coupled qubits with transverse global control, Phys. Rev. A 106, 052613 (2022).
  57. E. Lötstedt and K. Yamanouchi, Comparison of encoding schemes for quantum computing of S>1/2 spin chains, Phys. Rev. A 111, 062416 (2025).
  58. T. Haase, G. Alber, and V. M. Stojanović, Conversion from W to Greenberger-Horne-Zeilinger states in the Rydberg-blockade regime of neutral-atom systems: Dynamical-symmetry-based approach, Phys. Rev. A 103, 032427 (2021).
  59. J. K. Nauth and V. M. Stojanović, Quantum-brachistochrone approach to the conversion from W to Greenberger-Horne-Zeilinger states for Rydberg-atom qubits, Phys. Rev. A 106, 032605 (2022).
  60. H. Sharma and U. T. Bhosale, Exact solvability of entanglement for arbitrary initial state in an infinite-range Floquet system, Ann. Phys. (NY) 486, 170327 (2026).
  61. G. Toth, Detection of multipartite entanglement in the vicinity of symmetric Dicke states, J. Opt. Soc. Am. B 24, 275 (2007).
  62. W. Pfeifer, The Lie Algebras su(N): An Introduction (Birkhäuser, Basel, 2003).
  63. See, e.g., S. Lorenzo, T. J. G. Apollaro, A. Sindona, and F. Plastina, Quantum-state transfer via resonant tunneling through local-field-induced barriers, Phys. Rev. A 87, 042313 (2013).
  64. M. Veldhorst, H. G. J. Eenink, C. H. Yang, and A. S. Dzurak, Silicon CMOS architecture for a spin-based quantum computer, Nat. Commun. 8, 1766 (2017).
  65. See, e.g., I. Hansen, A. E. Seedhouse, A. Saraiva, A. Laucht, A. S. Dzurak, and C. H. Yang, Pulse engineering of a global field for robust and universal quantum computation, Phys. Rev. A 104, 062415 (2021).
  66. H. C. George, M. T. Madzik, E. M. Henry, A. J. Wagner, M. M. Islam, F. Borjans, E. J. Connors, J. Corrigan, M. Curry, M. K. Harper, et al., 12-spin-qubit arrays fabricated on a 300 mm semiconductor manufacturing line, Nano Lett. 25, 793 (2025).
  67. R. Aiudi, J. Despres, R. Menta, A. Abedi, G. Menichetti, V. Giovannetti, M. Polini, and F. Caravelli, Overcoming disorder in superconducting globally driven quantum computing, Phys. Rev. A 113, 012616 (2026).
  68. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  69. J. Chen, Y. Zhou, J. Bian, J. Li, and X. Peng, Subspace controllability of symmetric spin networks, Phys. Rev. A 102, 032602 (2020).
  70. V. Evangelakos, E. Paspalakis, and D. Stefanatos, Fast charging of an Ising-spin-pair quantum battery using optimal control, Phys. Rev. A 110, 052601 (2024).
  71. V. Evangelakos, E. Paspalakis, and D. Stefanatos, Fast protocols for charging a three-spin-chain quantum battery, Sci. Rep. 15, 45626 (2025).
  72. H. Zhou, Q. Hu, Y. Chen, T. Wang, F. Jin, Y. Ji, J. Geng, and X. Peng, Fast conversion from W to Greenberger-Horne-Zeilinger states via inverse engineering, Phys. Rev. Appl. 25, 014056 (2026).
  73. V. M. Stojanović, A. Fedorov, A. Wallraff, and C. Bruder, Quantum-control approach to realizing a Toffoli gate in circuit QED, Phys. Rev. B 85, 054504 (2012).
  74. 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).
  75. M. H. Goerz, D. Basilewitsch, F. Gago-Encinas, M. G. Krauss, K. P. Horn, D. M. Reich, and C. Koch, Krotov: A Python implementation of Krotov's method for quantum optimal control, SciPost Phys. 7, 080 (2019).
  76. D. Lewis, R. Wiersema, and S. Bose, Quantum optimal control with geodesic pulse engineering, arXiv:2508.16029.
  77. T. Caneva, T. Calarco, and S. Montangero, Chopped random-basis quantum optimization, Phys. Rev. A 84, 022326 (2011).
  78. D. Lucarelli, Quantum optimal control via gradient ascent in function space and the time-bandwidth quantum speed limit, Phys. Rev. A 97, 062346 (2018).
  79. See, e.g., T. N. Kaldenbach, M. Heller, G. Alber, and V. M. Stojanović, Digital quantum simulation of scalar Yukawa coupling, Quantum Rep. 6, 366 (2024).
  80. Ch. Tsitouras, Runge–Kutta pairs of order 5(4) satisfying only the first column simplifying assumption, Comput. Math. Appl. 62, 770 (2011).
  81. V. M. Stojanović and M. Vanević, Quantum-entanglement aspects of polaron systems, Phys. Rev. B 78, 214301 (2008).
  82. See, e.g., V. M. Stojanović, Entanglement-spectrum characterization of ground-state nonanalyticities in coupled excitation-phonon models, Phys. Rev. B 101, 134301 (2020).
  83. A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, et al., Generation and manipulation of Schrödinger cat states in Rydberg atom arrays, Science 365, 570 (2019).
  84. T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, Arbitrary Dicke-state control of symmetric Rydberg ensembles, Phys. Rev. Lett. 117, 213601 (2016).
  85. C. Arenz and H. Rabitz, Controlling qubit networks in polynomial time, Phys. Rev. Lett. 120, 220503 (2018).
  86. A. Nickl, N. D. Stuyck, P. Steinacker, J. D. Cifuentes, S. Serrano, M. Feng, E. Vahapoglu, F. E. Hudson, K. W. Chan, S. Kubicek, et al., Eight-qubit operation of a 300 mm SiMOS foundry-fabricated device, arXiv:2512.10174.
  87. For a review, see S. Deffner and S. Campbell, Quantum speed limits: From Heisenberg's uncertainty principle to optimal quantum control, J. Phys. A: Math. Theor. 50, 453001 (2017).
  88. T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, and G. E. Santoro, Optimal control at the quantum speed limit, Phys. Rev. Lett. 103, 240501 (2009).
  89. V. M. Stojanović and I. Salom, Quantum dynamics of the small-polaron formation in a superconducting analog simulator, Phys. Rev. B 99, 134308 (2019).
  90. M. Pioro-Ladriere, T. Obata, Y. Tokura, Y.-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Electrically driven single-electron spin resonance in a slanting Zeeman field, Nat. Phys. 4, 776 (2008).

Outline

Information

Phys. Rev. Research 8, 033095– Published 23 July, 2026
  • Received 29 January 2026
  • Accepted 22 June 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