- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

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

Asymptotically Solvable Quantum Circuits

Samuel H. Pickering and Bruno Bertini*

  • *Contact author: b.bertini@bham.ac.uk
PDF

PRX Quantum 7, 043001 – Published 1 October, 2026

DOI: https://doi.org/10.1103/7g63-nhl9

Abstract

The discovery of chaotic quantum circuits with (partially) solvable dynamics has played a key role in our understanding of non-equilibrium quantum matter and, at the same time, has helped the development of concrete platforms for quantum computation. It was shown that solvability does not prevent the generation of chaotic dynamics, however, it imposes nontrivial constraints on the generated correlations. A natural question is then whether it is possible to gain insight into the generic case despite the latter being very hard to access. To address this question here we introduce a family of “asymptotically solvable” quantum circuits where the solvability constraints only affect correlations on length scales beyond a tunable threshold. This means that their dynamics are only solvable for long enough times: for times shorter than the threshold they are generic. We show this by computing both their dynamical correlations on the equilibrium (infinite temperature) state and their thermalization dynamics following quantum quenches from compatible (asymptotically solvable) non-equilibrium initial states. The class of systems we introduce is generically ergodic but contains a non-interacting point, which we use to provide exact analytical results, complementing those of numerical experiments, on the non-solvable early time regime.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Quantum mechanics is arguably one of humanity’s highest achievements and provides the foundation to most of the theoretical and technological progress of the last 100 years. At the same time, however, it is also remarkably complicated. In fact, describing large collections of quantum particles interacting together is still an insurmountable barrier for today’s scientists. Yet, quantum many-body systems are ubiquitous in Nature, and understanding their dynamics is crucial for both humanity’s scientific progress and for exploiting their potential technological applications, for example, to build the much awaited quantum computers. The discovery of interacting quantum systems with solvable dynamics, solvable quantum circuits, has brought new hope. This was achieved by imposing certain conditions on the allowed interactions that, remarkably, did not make them trivial. A natural question is whether these constraints can be systematically lifted to get back to generic dynamics. Here we address it by introducing a family of “asymptotically solvable” quantum circuits where the solvability constraints only affect correlations on length scales beyond a tunable threshold.

Article Text

References (98)

  1. J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nat. Phys. 11, 124 (2015).
  2. P. Calabrese, F. H. L. Essler, and G. Mussardo, Introduction to ‘quantum integrability in out of equilibrium systems’, J. Stat. Mech. 2016, 064001
  3. M. Serbyn, D. A. Abanin, and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021).
  4. A. Bastianello, B. Bertini, B. Doyon, and R. Vasseur, Introduction to the special issue on emergent hydrodynamics in integrable many-body systems, J. Stat. Mech. 2022, 014001
  5. P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 2005, P04010
  6. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, J. High Energy Phys. 09 (2007) 120
  7. Y. Sekino and L. Susskind, Fast scramblers, J. High Energy Phys. 10 (2008) 065
  8. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  9. S. H. Shenker and D. Stanford, Multiple shocks, J. High Energy Phys. 2014 (2014) 46.
  10. A. Kitaev, A simple model of quantum holography, in KITP Program: Entanglement in Strongly-Correlated Quantum Matter (2015).
  11. P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, J. High Energy Phys. 2 (2016) 1
  12. B. Bertini, K. Klobas, V. Alba, G. Lagnese, and P. Calabrese, Growth of Rényi entropies in interacting integrable models and the breakdown of the quasiparticle picture, Phys. Rev. X 12, 031016 (2022).
  13. K. Sacha and J. Zakrzewski, Time crystals: A review, Rep. Prog. Phys. 81, 016401 (2017).
  14. V. Khemani, R. Moessner, and S. L. Sondhi, A Brief History of Time Crystals, arXiv:1910.10745.
  15. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  16. Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  17. A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, in Entanglement in Spin Chains: From Theory to Quantum Technology Applications, edited by A. Bayat, S. Bose, and H. Johannesson (Springer International Publishing, Cham, 2022), pp. 211–249.
  18. M. P. A. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023).
  19. B. Bertini, P. W. Claeys, and T. Prosen, Exactly solvable many-body dynamics from space-time duality, arXiv:2505.11489.
  20. B. Bertini, Non-equilibrium quantum many-body physics with quantum circuits, arXiv:2601.22375.
  21. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  22. B. Bertini, P. Kos, and T. Prosen, Exact correlation functions for dual-unitary lattice models in 1+1 dimensions, Phys. Rev. Lett. 123, 210601 (2019).
  23. X.-H. Yu, Z. Wang, and P. Kos, Hierarchical generalization of dual unitarity, Quantum 8, 1260 (2024).
  24. K. Klobas, B. Bertini, and L. Piroli, Exact thermalization dynamics in the “rule 54” quantum cellular automaton, Phys. Rev. Lett. 126, 160602 (2021).
  25. B. Bertini, P. Kos, and T. c. v. Prosen, Exact spectral statistics in strongly localized circuits, Phys. Rev. B 105, 165142 (2022).
  26. K. Klobas, Exact subsystem dynamics in the deterministic floquet PXP model (To be published).
  27. B. Bertini, P. Kos, and T. Prosen, Entanglement spreading in a minimal model of maximal many-body quantum chaos, Phys. Rev. X 9, 021033 (2019).
  28. S. Gopalakrishnan and A. Lamacraft, Unitary circuits of finite depth and infinite width from quantum channels, Phys. Rev. B 100, 064309 (2019).
  29. L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, Exact dynamics in dual-unitary quantum circuits, Phys. Rev. B 101, 094304 (2020).
  30. T. Zhou and A. W. Harrow, Maximal entanglement velocity implies dual unitarity, Phys. Rev. B 106, L201104 (2022).
  31. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  32. C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018).
  33. P. W. Claeys and A. Lamacraft, Maximum velocity quantum circuits, Phys. Rev. Res. 2, 033032 (2020).
  34. W. W. Ho and S. Choi, Exact emergent quantum state designs from quantum chaotic dynamics, Phys. Rev. Lett. 128, 060601 (2022).
  35. M. Ippoliti and W. W. Ho, Dynamical purification and the emergence of quantum state designs from the projected ensemble, PRX Quantum 4, 030322 (2023).
  36. A. Chan, A. De Luca, and J. T. Chalker, Solution of a minimal model for many-body quantum chaos, Phys. Rev. X 8, 041019 (2018).
  37. A. Chan, A. De Luca, and J. T. Chalker, Spectral statistics in spatially extended chaotic quantum many-body systems, Phys. Rev. Lett. 121, 060601 (2018).
  38. B. Bertini, P. Kos, and T. c. v. Prosen, Exact spectral form factor in a minimal model of many-body quantum chaos, Phys. Rev. Lett. 121, 264101 (2018).
  39. A. J. Friedman, A. Chan, A. De Luca, and J. T. Chalker, Spectral statistics and many-body quantum chaos with conserved charge, Phys. Rev. Lett. 123, 210603 (2019).
  40. B. Bertini, P. Kos, and T. Prosen, Random matrix spectral form factor of dual-unitary quantum circuits, Commun. Math. Phys. 387, 597 (2021).
  41. F. Fritzsch and T. Prosen, Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions, Phys. Rev. E 103, 062133 (2021).
  42. F. Fritzsch, M. F. I. Kieler, and A. Bäcker, Eigenstate correlations in dual-unitary quantum circuits: Partial spectral form factor, Quantum 9, 1709 (2025).
  43. X. Mi, P. Roushan, C. Quintana, S. Mandrá, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
  44. E. Chertkov, J. Bohnet, D. Francois, J. Gaebler, D. Gresh, A. Hankin, K. Lee, D. Hayes, B. Neyenhuis, R. Stutz, A. C. Potter, and M. Foss-Feig, Holographic dynamics simulations with a trapped-ion quantum computer, Nat. Phys. 18, 1074 (2022).
  45. L. E. Fischer, M. Leahy, A. Eddins, N. Keenan, D. Ferracin, M. A. C. Rossi, Y. Kim, A. He, F. Pietracaprina, B. Sokolov, S. Dooley, Z. Zimborás, F. Tacchino, S. Maniscalco, J. Goold, G. García-Pérez, I. Tavernelli, A. Kandala, and S. N. Filippov, Dynamical simulations of many-body quantum chaos on a quantum computer, arXiv:2411.00765.
  46. B. Bertini, Mitigated chaos, Nat. Phys. 22, 182 (2026).
  47. T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev. B 99, 174205 (2019).
  48. Y. Kasim and T. Prosen, Dual unitary circuits in random geometries, J. Phys. A 56, 025003 (2023).
  49. P. W. Claeys and A. Lamacraft, Operator dynamics and entanglement in space-time dual Hadamard lattices, J. Phys. A 57, 405301 (2024).
  50. P. W. Claeys, A. Lamacraft, and J. Vicary, From dual-unitary to biunitary: A 2-categorical model for exactly-solvable many-body quantum dynamics, J. Phys. A 57, 335301 (2024).
  51. M. A. Rampp, S. A. Rather, and P. W. Claeys, Solvable quantum circuits from spacetime lattices, arXiv:2512.15871.
  52. A. Foligno, P. Kos, and B. Bertini, Quantum information spreading in generalized dual-unitary circuits, Phys. Rev. Lett. 132, 250402 (2024).
  53. B. Bertini, C. De Fazio, J. P. Garrahan, and K. Klobas, Exact quench dynamics of the floquet quantum east model at the deterministic point, Phys. Rev. Lett. 132, 120402 (2024).

  54. Or better, isometric property, see the discussion in Sec. 1a.

  55. T. Prosen, Many-body quantum chaos and dual-unitarity round-a-face, Chaos Interdiscip. J. Nonlinear Sci. 31, 093101 (2021).
  56. C. Jonay, V. Khemani, and M. Ippoliti, Triunitary quantum circuits, Phys. Rev. Res. 3, 043046 (2021).
  57. R. Suzuki, K. Mitarai, and K. Fujii, Computational power of one- and two-dimensional dual-unitary quantum circuits, Quantum 6, 631 (2022).
  58. M. A. Rampp, S. A. Rather, and P. W. Claeys, Geometric constructions of generalized dual-unitary circuits from biunitarity, SciPost Phys. 18, 182 (2025).
  59. R. M. Milbradt, L. Scheller, C. Aßmus, and C. B. Mendl, Ternary unitary quantum lattice models and circuits in 2+1 dimensions, Phys. Rev. Lett. 130, 090601 (2023).
  60. M. Mestyán, B. Pozsgay, and I. M. Wanless, Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states, SciPost Phys. 16, 010 (2024).

  61. Here we use the convention that matrix multiplication goes from bottom to top, i.e. AB is represented with the symbol for A above the one for B.

  62. A. Lerose, M. Sonner, and D. A. Abanin, Influence matrix approach to many-body floquet dynamics, Phys. Rev. X 11, 021040 (2021).
  63. B. Bertini and L. Piroli, Scrambling in random unitary circuits: Exact results, Phys. Rev. B 102, 064305 (2020).
  64. N. Dowling, P. Kos, and K. Modi, Scrambling is necessary but not sufficient for chaos, Phys. Rev. Lett. 131, 180403 (2023).
  65. A. Foligno and B. Bertini, Growth of entanglement of generic states under dual-unitary dynamics, Phys. Rev. B 107, 174311 (2023).
  66. G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D. A. Abanin, and L. Piroli, Temporal entanglement, quasiparticles, and the role of interactions, Phys. Rev. Lett. 128, 220401 (2022).
  67. A. Foligno, T. Zhou, and B. Bertini, Temporal entanglement in chaotic quantum circuits, Phys. Rev. X 13, 041008 (2023).
  68. B. Bertini, P. Kos, and T. Prosen, Operator entanglement in local quantum circuits I: Chaotic dual-unitary circuits, SciPost Phys. 8, 67 (2020).
  69. B. Bertini, P. Kos, and T. Prosen, Operator entanglement in local quantum circuits II: Solitons in chains of qubits, Scipost Phys. 8, 68 (2020).
  70. I. Reid and B. Bertini, Entanglement barriers in dual-unitary circuits, Phys. Rev. B 104, 014301 (2021).
  71. P. W. Claeys and A. Lamacraft, Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics, Quantum 6, 738 (2022).
  72. O. Breach, B. Placke, P. W. Claeys, and S. A. Parameswaran, Solvable quantum circuits in tree+1 dimensions, arXiv:2503.20927.
  73. M. C. Bañuls, M. B. Hastings, F. Verstraete, and J. I. Cirac, Matrix product states for dynamical simulation of infinite chains, Phys. Rev. Lett. 102, 240603 (2009).
  74. H. Casini, H. Liu, and M. Mezei, Spread of entanglement and causality, J. High Energy Phys. 2016, 77 (2016).
  75. K. Klobas and B. Bertini, Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54, SciPost Phys. 11, 106 (2021).
  76. P. Zanardi, C. Zalka, and L. Faoro, Entangling power of quantum evolutions, Phys. Rev. A 62, 030301 (2000).
  77. S. Aravinda, S. A. Rather, and A. Lakshminarayan, From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy, Phys. Rev. Res. 3, 043034 (2021).
  78. T. Prosen, General relation between quantum ergodicity and fidelity of quantum dynamics, Phys. Rev. E 65, 036208 (2002).
  79. T. Prosen, Chaos and complexity of quantum motion, J. Phys. A 40, 7881 (2007).
  80. C. Jonay, D. A. Huse, and A. Nahum, Coarse-grained dynamics of operator and state entanglement, arXiv:1803.00089.
  81. T. Zhou and A. Nahum, Entanglement membrane in chaotic many-body systems, Phys. Rev. X 10, 031066 (2020).
  82. M. Akila, D. Waltner, B. Gutkin, and T. Guhr, Particle-time duality in the kicked Ising spin chain, J. Phys. A 49, 375101 (2016).
  83. P. Kos, T. Prosen, and B. Bertini, Thermalization dynamics and spectral statistics of extended systems with thermalizing boundaries, Phys. Rev. B 104, 214303 (2021).
  84. A. Foligno, P. Calabrese, and B. Bertini, Nonequilibrium dynamics of charged dual-unitary circuits, PRX Quantum 6, 010324 (2025).
  85. M. A. Rampp, S. A. Rather, and P. W. Claeys, Entanglement membrane in exactly solvable lattice models, Phys. Rev. Res. 6, 033271 (2024).
  86. G. M. Sommers, S. Gopalakrishnan, M. J. Gullans, and D. A. Huse, Zero-temperature entanglement membranes in quantum circuits, Phys. Rev. B 110, 064311 (2024).
  87. R. Modak, V. Alba, and P. Calabrese, Entanglement revivals as a probe of scrambling in finite quantum systems, J. Stat. Mech. Theory Exp. 2020, 083110 (2020).
  88. L. Capizzi and V. Eisler, Entanglement evolution after a global quench across a conformal defect, SciPost Phys. 14, 070 (2023).
  89. Ž. Krajnik, E. Ilievski, and T. Prosen, Absence of normal fluctuations in an integrable magnet, Phys. Rev. Lett. 128, 090604 (2022).
  90. J. Feldmeier, W. Witczak-Krempa, and M. Knap, Emergent tracer dynamics in constrained quantum systems, Phys. Rev. B 106, 094303 (2022).
  91. J. De Nardis, S. Gopalakrishnan, and R. Vasseur, Nonlinear fluctuating hydrodynamics for Kardar-Parisi-Zhang scaling in isotropic spin chains, Phys. Rev. Lett. 131, 197102 (2023).
  92. S. Gopalakrishnan, A. Morningstar, R. Vasseur, and V. Khemani, Distinct universality classes of diffusive transport from full counting statistics, Phys. Rev. B 109, 024417 (2024).
  93. Ž. Krajnik, J. Schmidt, E. Ilievski, and T. Prosen, Dynamical criticality of magnetization transfer in integrable spin chains, Phys. Rev. Lett. 132, 017101 (2024).
  94. E. Rosenberg, T. I. Andersen, R. Samajdar, A. Petukhov, J. C. Hoke, D. Abanin, A. Bengtsson, I. K. Drozdov, C. Erickson et al., Dynamics of magnetization at infinite temperature in a Heisenberg spin chain, Science 384, 48 (2024).
  95. D. J. Thouless, Maximum metallic resistance in thin wires, Phys. Rev. Lett. 39, 1167 (1977).
  96. S. H. Pickering and B. Bertini, Simulation data for “asymptotically solvable quantum circuits”, (2026).
  97. P. Fendley, Free fermions in disguise, J. Phys. A 52, 335002 (2019).
  98. A. Chapman and S. T. Flammia, Characterization of solvable spin models via graph invariants, Quantum 4, 278 (2020).

Outline

Information

PRX Quantum 7, 043001– Published 1 October, 2026
  • Received 20 March 2026
  • Revised 3 July 2026
  • Accepted 31 July 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