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  • Open Access

Asymptotic Exceptional Steady States in Dissipative Dynamics

Yu-Min Hu1 and Jan Carl Budich1,2,*

  • *Contact author: jan.budich@tu-dresden.de
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Phys. Rev. Lett. 135, 250402 – Published 17 December, 2025

DOI: https://doi.org/10.1103/6lkq-8626

Abstract

Spectral degeneracies in Liouvillian generators of dissipative dynamics generically occur as exceptional points, where the corresponding non-Hermitian operator becomes nondiagonalizable. Steady states, i.e., zero modes of Liouvillians, are considered a fundamental exception to this rule since a no-go theorem excludes nondiagonalizable degeneracies there. Here, we demonstrate that the crucial issue of diverging timescales in dissipative state preparation is largely tantamount to an asymptotic approach toward the forbidden scenario of an exceptional steady state in the thermodynamic limit. With case studies ranging from NP-complete satisfiability problems encoded in a quantum master equation to the dissipative preparation of a symmetry protected topological phase, we reveal the close relation between the computational complexity of the problem at hand, and the finite size scaling toward the exceptional steady state, exemplifying both exponential and polynomial scaling. Formally treating the weight W of quantum jumps in the Lindblad master equation as a parameter, we show that exceptional steady states at the physical value W=1 may be understood as a critical point hallmarking the onset of dynamical instability.

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References (79)

  1. S. A. Cook, The complexity of theorem-proving procedures, in Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC ’71 (Association for Computing Machinery, New York, NY, USA, 1971), pp. 151–158.
  2. R. M. Karp, Reducibility among combinatorial problems, in 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art (Springer, New York, 2009), pp. 219–241.
  3. M. Mézard and R. Zecchina, Random k-satisfiability problem: From an analytic solution to an efficient algorithm, Phys. Rev. E 66, 056126 (2002).
  4. A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and Quantum Computation, Vol. 47 (American Mathematical Society, Providence, 2002).
  5. J. Kempe, A. Kitaev, and O. Regev, The complexity of the local Hamiltonian problem, SIAM J. Comput. 35, 1070 (2006).
  6. D. Aharonov, W. Van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, Adiabatic quantum computation is equivalent to standard quantum computation, SIAM Rev. 50, 755 (2008).
  7. I. Buluta and F. Nori, Quantum simulators, Science 326, 108 (2009).
  8. J. I. Cirac and P. Zoller, Goals and opportunities in quantum simulation, Nat. Phys. 8, 264 (2012).
  9. I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  10. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature (London) 607, 667 (2022).
  11. E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem, Science 292, 472 (2001).
  12. A. Das and B. K. Chakrabarti, Colloquium: Quantum annealing and analog quantum computation, Rev. Mod. Phys. 80, 1061 (2008).
  13. T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  14. A. P. Young, S. Knysh, and V. N. Smelyanskiy, First-order phase transition in the quantum adiabatic algorithm, Phys. Rev. Lett. 104, 020502 (2010).
  15. T. Jörg, F. Krzakala, G. Semerjian, and F. Zamponi, First-order transitions and the performance of quantum algorithms in random optimization problems, Phys. Rev. Lett. 104, 207206 (2010).
  16. B. Altshuler, H. Krovi, and J. Roland, Anderson localization makes adiabatic quantum optimization fail, Proc. Natl. Acad. Sci. U.S.A. 107, 12446 (2010).
  17. M. H. S. Amin and V. Choi, First-order quantum phase transition in adiabatic quantum computation, Phys. Rev. A 80, 062326 (2009).
  18. N. G. Dickson and M. H. S. Amin, Does adiabatic quantum optimization fail for NP-complete problems?, Phys. Rev. Lett. 106, 050502 (2011).
  19. E. Farhi, J. Goldstone, D. Gosset, S. Gutmann, H. B. Meyer, and P. Shor, Quantum adiabatic algorithms, small gaps, and different paths, Quantum Inf. Comput. 11, 0181 (2011).
  20. M. Werner, A. García-Sáez, and M. P. Estarellas, Bounding first-order quantum phase transitions in adiabatic quantum computing, Phys. Rev. Res. 5, 043236 (2023).
  21. B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller, Preparation of entangled states by quantum Markov processes, Phys. Rev. A 78, 042307 (2008).
  22. S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Büchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008).
  23. F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009).
  24. S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topology by dissipation in atomic quantum wires, Nat. Phys. 7, 971 (2011).
  25. C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. İmamoğlu, P. Zoller, and S. Diehl, Topology by dissipation, New J. Phys. 15, 085001 (2013).
  26. J. C. Budich, P. Zoller, and S. Diehl, Dissipative preparation of Chern insulators, Phys. Rev. A 91, 042117 (2015).
  27. M. Goldstein, Dissipation-induced topological insulators: A no-go theorem and a recipe, SciPost Phys. 7, 067 (2019).
  28. Z. Liu, E. J. Bergholtz, and J. C. Budich, Dissipative preparation of fractional Chern insulators, Phys. Rev. Res. 3, 043119 (2021).
  29. F. Yang, P. Molignini, and E. J. Bergholtz, Dissipative boundary state preparation, Phys. Rev. Res. 5, 043229 (2023).
  30. S. Bandyopadhyay and A. Dutta, Dissipative preparation of many-body floquet Chern insulators, Phys. Rev. B 102, 184302 (2020).
  31. L. Zhou, S. Choi, and M. D. Lukin, Symmetry-protected dissipative preparation of matrix product states, Phys. Rev. A 104, 032418 (2021).
  32. Y. Lin, J. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. Sørensen, D. Leibfried, and D. J. Wineland, Dissipative production of a maximally entangled steady state of two quantum bits, Nature (London) 504, 415 (2013).
  33. F. Reiter, D. Reeb, and A. S. Sørensen, Scalable dissipative preparation of many-body entanglement, Phys. Rev. Lett. 117, 040501 (2016).
  34. M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Dissipative preparation of entanglement in optical cavities, Phys. Rev. Lett. 106, 090502 (2011).
  35. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n-level systems, J. Math. Phys. (N.Y.) 17, 821 (1976).
  36. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  37. J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).
  38. M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Rev. Mod. Phys. 70, 101 (1998).
  39. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  40. J. P. Garrahan and I. Lesanovsky, Thermodynamics of quantum jump trajectories, Phys. Rev. Lett. 104, 160601 (2010).
  41. F. Carollo, R. L. Jack, and J. P. Garrahan, Unraveling the large deviation statistics of markovian open quantum systems, Phys. Rev. Lett. 122, 130605 (2019).
  42. See Supplemental Material available at http://link.aps.org/supplemental/10.1103/6lkq-8626 for details.
  43. W. Heiss, Exceptional points of non-Hermitian operators, J. Phys. A 37, 2455 (2004).
  44. M. V. Berry, Physics of non Hermitian degeneracies, Czech. J. Phys. 54, 1039 (2004).
  45. W. D. Heiss, The physics of exceptional points, J. Phys. A 45, 444016 (2012).
  46. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  47. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  48. M.-A. Miri and A. Alu, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  49. F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019).
  50. I. I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between PT and anti-PT symmetries, Phys. Rev. A 102, 033715 (2020).
  51. F. Minganti, A. Miranowicz, R. W. Chhajlany, I. I. Arkhipov, and F. Nori, Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories, Phys. Rev. A 101, 062112 (2020).
  52. S. Khandelwal, N. Brunner, and G. Haack, Signatures of Liouvillian exceptional points in a quantum thermal machine, PRX Quantum 2, 040346 (2021).
  53. W. Chen, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum jumps in the non-Hermitian dynamics of a superconducting qubit, Phys. Rev. Lett. 127, 140504 (2021).
  54. J.-W. Zhang, J.-Q. Zhang, G.-Y. Ding, J.-C. Li, J.-T. Bu, B. Wang, L.-L. Yan, S.-L. Su, L. Chen, F. Nori et al., Dynamical control of quantum heat engines using exceptional points, Nat. Commun. 13, 6225 (2022).
  55. W. Chen, M. Abbasi, B. Ha, S. Erdamar, Y. N. Joglekar, and K. W. Murch, Decoherence-induced exceptional points in a dissipative superconducting qubit, Phys. Rev. Lett. 128, 110402 (2022).
  56. Y.-L. Zhou, X.-D. Yu, C.-W. Wu, X.-Q. Li, J. Zhang, W. Li, and P.-X. Chen, Accelerating relaxation through Liouvillian exceptional point, Phys. Rev. Res. 5, 043036 (2023).
  57. J.-T. Bu, J.-Q. Zhang, G.-Y. Ding, J.-C. Li, J.-W. Zhang, B. Wang, W.-Q. Ding, W.-F. Yuan, L. Chen, i. m. c. K. Özdemir, F. Zhou, H. Jing, and M. Feng, Enhancement of quantum heat engine by encircling a liouvillian exceptional point, Phys. Rev. Lett. 130, 110402 (2023).

  58. EPs are more abundant than diagonalizable degeneracies for generic NH matrices. Specifically, for complex 2×2 matrices with 8 real parameters, EPs form a 6-dimensional manifold compared to a 2-dimensional manifold of diagonalizable degeneracies.

  59. F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spectral theory of Liouvillians for dissipative phase transitions, Phys. Rev. A 98, 042118 (2018).
  60. D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-free subspaces for quantum computation, Phys. Rev. Lett. 81, 2594 (1998).
  61. S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Inf. 5, 75 (2019).
  62. H. Nishi, T. Kosugi, and Y.-i. Matsushita, Implementation of quantum imaginary-time evolution method on NISQ devices by introducing nonlocal approximation, npj Quantum Inf. 7, 85 (2021).
  63. T. Mori and T. Shirai, Resolving a discrepancy between liouvillian gap and relaxation time in boundary-dissipated quantum many-body systems, Phys. Rev. Lett. 125, 230604 (2020).
  64. T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Liouvillian skin effect: Slowing down of relaxation processes without gap closing, Phys. Rev. Lett. 127, 070402 (2021).

  65. Our Letter focuses on the spectrum near the steady state which can be generically assumed to be diagonalizable. A nondiagonalizable Liouvillian can be treated similarly by replacing the eigenvalues and eigenstates of decaying modes with those of the corresponding Jordan blocks.

  66. The effective description LWeff in Eq. (4) indicates that an open quantum system at W=1 cannot approach the AESS when only one of the limits λ→0 and |ν|2→1 is achieved. Examples of those cases are presented in the SM [42].

  67. A. Lucas, Ising formulations of many NP problems, Front. Phys. 2, 5 (2014).

  68. The steady state corresponding to a unsatisfiable 3SAT instance is generally a complicated mixed state.

  69. W. Barthel, A. K. Hartmann, M. Leone, F. Ricci-Tersenghi, M. Weigt, and R. Zecchina, Hiding solutions in random satisfiability problems: A statistical mechanics approach, Phys. Rev. Lett. 88, 188701 (2002).

  70. When λ0 and λ1 in Fig. 2 cross at W=Wc, we keep denoting the eigenvalue of the solution state |ψsol⟩⟨ψsol| with λ0 for simplicity. This keeps Eq. (4) valid at both W>Wc and W<Wc without exchanging basis vectors.

  71. C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  72. C. M. Bender, M. Berry, and A. Mandilara, Generalized PT symmetry and real spectra, J. Phys. A 35, L467 (2002).
  73. A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. (N.Y.) 43, 205 (2002).
  74. A. Mostafazadeh, Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. (N.Y.) 43, 2814 (2002).
  75. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  76. Y. Wang, K. Snizhko, A. Romito, Y. Gefen, and K. Murch, Dissipative preparation and stabilization of many-body quantum states in a superconducting qutrit array, Phys. Rev. A 108, 013712 (2023).
  77. F. Carollo, A. Lasanta, and I. Lesanovsky, Exponentially accelerated approach to stationarity in markovian open quantum systems through the Mpemba effect, Phys. Rev. Lett. 127, 060401 (2021).
  78. X.-K. Gu, L.-Z. Tan, F. Nori, and J. Q. You, Exploring dynamics of open quantum systems in naturally inaccessible regimes, arXiv:2503.06946.
  79. Zenodo data archive at: 10.5281/zenodo.17661045.

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Phys. Rev. Lett. 135, 250402– Published 17 December, 2025
  • Received 7 April 2025
  • Revised 5 August 2025
  • Accepted 17 November 2025
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