- 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
  • Letter
  • Open Access

Solvable entanglement dynamics in quantum circuits with generalized space-time duality

Chuan Liu1,* and Wen Wei Ho1,2,†

  • *Contact author: c_liu@u.nus.edu
  • †Contact author: wenweiho@nus.edu.sg
PDF

Phys. Rev. Research 7, L012011 – Published 17 January, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L012011

Abstract

We study the nonequilibrium dynamics of kicked Ising models in 1 + 1 dimensions which have interactions alternating between odd and even bonds in time. These models can be understood as quantum circuits tiling space-time with the generalized space-time dual properties of triunitarity (three “arrows of time”) at the global level, and also second-level dual unitarity at the local level, which constrains the behavior of pairs of local gates underlying the circuit under a space-time rotation. We identify a broad class of initial product states wherein the effect of the environment on a small subsystem can be exactly represented by influence matrices with simple Markovian structures, resulting in the subsystem's full dynamics being efficiently computable. We further find additional conditions under which the dynamics of entanglement can be solved for all times, yielding rich phenomenology ranging from linear growth at half the maximal speed allowed by locality, followed by saturation to maximum entropy (i.e., thermalization to infinite temperature), to entanglement growth with saturation to extensive but submaximal entropy. Intriguingly, for certain parameter regimes, we find a nonchaotic class of dynamics which is neither integrable nor Clifford, exemplified by nonzero operator entanglement growth but with a spectral form factor which exhibits large, apparently time-quasiperiodic revivals.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (68)

  1. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  2. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  3. L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  4. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  5. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, J. High Energy Phys. 09 (2007) 120.
  6. S. H. Shenker and D. Stanford, Black holes and the butterfly effect, J. High Energy Phys. 03 (2014) 067.
  7. P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, J. High Energy Phys. 02 (2016) 004.
  8. D. A. Roberts and B. Yoshida, Chaos and complexity by design, J. High Energy Phys. 04 (2017) 121.
  9. M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, J. High Energy Phys. 05 (2017) 065.
  10. K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Verified quantum information scrambling, Nature (London) 567, 61 (2019).
  11. H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys. 86, 779 (2014).
  12. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  13. A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces, J. Stat. Mech. (2004) P04005.
  14. S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. 411, 167998 (2019).
  15. K. Hémery, F. Pollmann, and D. J. Luitz, Matrix product states approaches to operator spreading in ergodic quantum systems, Phys. Rev. B 100, 104303 (2019).
  16. V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Natl. Acad. Sci. 114, 7947 (2017).
  17. F. H. L. Essler and M. Fagotti, Quench dynamics and relaxation in isolated integrable quantum spin chains, J. Stat. Mech. (2016) 064002.
  18. P. Calabrese, Entanglement spreading in non-equilibrium integrable systems, SciPost Physics Lecture Notes 20 (2020).
  19. K. Klobas, B. Bertini, and L. Piroli, Exact thermalization dynamics in the “rule 54” quantum cellular automaton, Phys. Rev. Lett. 126, 160602 (2021).
  20. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, Cambridge, 2000).
  21. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  22. T. Farshi, J. Richter, D. Toniolo, A. Pal, and L. Masanes, Absence of localization in two-dimensional Clifford circuits, PRX Quantum 4, 030302 (2023).
  23. J. Richter, O. Lunt, and A. Pal, Transport and entanglement growth in long-range random Clifford circuits, Phys. Rev. Res. 5, L012031 (2023).
  24. B. Bertini, P. Kos, and T. Prosen, Exact spectral form factor in a minimal model of many-body quantum chaos, Phys. Rev. Lett. 121, 264101 (2018).
  25. S. Gopalakrishnan and A. Lamacraft, Unitary circuits of finite depth and infinite width from quantum channels, Phys. Rev. B 100, 064309 (2019).
  26. S. A. Rather, S. Aravinda, and A. Lakshminarayan, Creating ensembles of dual unitary and maximally entangling quantum evolutions, Phys. Rev. Lett. 125, 070501 (2020).
  27. L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, Exact dynamics in dual-unitary quantum circuits, Phys. Rev. B 101, 094304 (2020).
  28. P. W. Claeys and A. Lamacraft, Ergodic and nonergodic dual-unitary quantum circuits with arbitrary local Hilbert space dimension, Phys. Rev. Lett. 126, 100603 (2021).
  29. A. Flack, B. Bertini, and T. Prosen, Statistics of the spectral form factor in the self-dual kicked Ising model, Phys. Rev. Res. 2, 043403 (2020).
  30. 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).
  31. M. B. Hastings and R. Mahajan, Connecting entanglement in time and space: Improving the folding algorithm, Phys. Rev. A 91, 032306 (2015).
  32. 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).
  33. A. Lerose, M. Sonner, and D. A. Abanin, Influence matrix approach to many-body Floquet dynamics, Phys. Rev. X 11, 021040 (2021).
  34. T.-C. Lu and T. Grover, Spacetime duality between localization transitions and measurement-induced transitions, PRX Quantum 2, 040319 (2021).
  35. M. Ippoliti, T. Rakovszky, and V. Khemani, Fractal, logarithmic, and volume-law entangled nonthermal steady states via spacetime duality, Phys. Rev. X 12, 011045 (2022).
  36. C. Jonay, V. Khemani, and M. Ippoliti, Triunitary quantum circuits, Phys. Rev. Res. 3, 043046 (2021).
  37. X.-H. Yu, Z. Wang, and P. Kos, Hierarchical generalization of dual unitarity, Quantum 8, 1260 (2024).
  38. V. Al. Osipov, N. Krieger, T. Guhr, and B. Gutkin, Local correlations in partially dual-unitary lattice models, Phys. Rev. B 109, 214302 (2024).
  39. W. W. Ho and S. Choi, Exact emergent quantum state designs from quantum chaotic dynamics, Phys. Rev. Lett. 128, 060601 (2022).
  40. J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernández, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Emergent quantum state designs from individual many-body wave functions, PRX Quantum 4, 010311 (2023).
  41. M. Ippoliti and W. W. Ho, Solvable model of deep thermalization with distinct design times, Quantum 6, 886 (2022).
  42. M. Ippoliti and W. W. Ho, Dynamical purification and the emergence of quantum state designs from the projected ensemble, PRX Quantum 4, 030322 (2023).
  43. H. Shrotriya and W. W. Ho, Nonlocality of deep thermalization, arXiv:2305.08437.
  44. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  45. 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).
  46. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  47. Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  48. S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum error correction in scrambling dynamics and measurement-induced phase transition, Phys. Rev. Lett. 125, 030505 (2020).
  49. Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020).
  50. M. J. Gullans and D. A. Huse, Scalable probes of measurement-induced criticality, Phys. Rev. Lett. 125, 070606 (2020).
  51. P. W. Claeys, M. Henry, J. Vicary, and A. Lamacraft, Exact dynamics in dual-unitary quantum circuits with projective measurements, Phys. Rev. Res. 4, 043212 (2022).
  52. B. Bertini, P. Kos, and T. Prosen, Localized dynamics in the Floquet quantum East model, Phys. Rev. Lett. 132, 080401 (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. N. Pancotti, G. Giudice, J. I. Cirac, J. P. Garrahan, and M. C. Bañuls, Quantum East model: Localization, nonthermal eigenstates, and slow dynamics, Phys. Rev. X 10, 021051 (2020).
  55. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.7.L012011 for details of tensor network notation and manipulation, as well as proofs of statements presented in the main text.
  56. A. J. Kollár, M. Fitzpatrick, and A. A. Houck, Hyperbolic lattices in circuit quantum electrodynamics, Nature (London) 571, 45 (2019).
  57. P. Bienias, I. Boettcher, R. Belyansky, A. J. Kollár, and A. V. Gorshkov, Circuit quantum electrodynamics in hyperbolic space: From photon bound states to frustrated spin models, Phys. Rev. Lett. 128, 013601 (2022).
  58. M. Sonner, A. Lerose, and D. A. Abanin, Influence functional of many-body systems: Temporal entanglement and matrix-product state representation, Ann. Phys. 435, 168677 (2021).
  59. 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).
  60. A. Lerose, M. Sonner, and D. A. Abanin, Scaling of temporal entanglement in proximity to integrability, Phys. Rev. B 104, 035137 (2021).
  61. A. Foligno, T. Zhou, and B. Bertini, Temporal entanglement in chaotic quantum circuits, Phys. Rev. X 13, 041008 (2023).
  62. A. Foligno, P. Kos, and B. Bertini, Quantum information spreading in generalized dual-unitary circuits, Phys. Rev. Lett. 132, 250402 (2024).
  63. T. Zhou and A. W. Harrow, Maximal entanglement velocity implies dual unitarity, Phys. Rev. B 106, L201104 (2022).
  64. M. L. Mehta, Random Matrices and the Statistical Theory of Spectra, 2nd ed. (Academic, New York, 1991).
  65. D. Gottesman, The Heisenberg representation of quantum computers, Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, edited by S. P. Corney, R. Delbourgo, and P. D. Jarvis (International Press, Cambridge, MA, 1999), pp. 32–43.
  66. D. Fattal, T. S. Cubitt, Y. Yamamoto, S. Bravyi, and I. L. Chuang, Entanglement in the stabilizer formalism, arXiv:quant-ph/0406168.
  67. J. Liu, Spectral form factors and late time quantum chaos, Phys. Rev. D 98, 086026 (2018).
  68. We note that each instance of the circuit from this ensemble has entanglement dynamics which is still analytically solvable.

Outline

Information

Phys. Rev. Research 7, L012011– Published 17 January, 2025
  • Received 5 January 2024
  • Revised 22 May 2024
  • Accepted 22 December 2024
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