- 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

Operator dynamics in k-Markov random circuits

Unnati Akhouri1,2, Pei-Jun Huang3, Elliott Rose1,2, and Sarah Shandera1,4,2,*

  • *Contact author: ses47@psu.edu
PDF

Phys. Rev. Research 8, 043016 – Published 5 October, 2026

DOI: https://doi.org/10.1103/p2cj-1kkl

Abstract

We demonstrate that k-Markov sequences of unitary gates provide low-cost handles to manipulate the rate and structure of information spreading compared to traditional random, 0-Markov, circuits. For swap gates and brickwork circuits, we use graph cover time to demonstrate how k-Markov processes can be used to control operator transport. With swap gates and the set of Clifford gates that can change operator weight, we show how k-Markov sequences can be used to manipulate scrambling time and generate novel structures of spatial-temporal correlations across a qubit network. We show that k-Markov circuits constructed from pswap gates at fixed angle are equivalent to standard brickwork circuits with pswap angle drawn from nonuniform distributions generated by the k-Markov process. In those circuits, the time evolution of the average Hamming weight and the space-time correlation structure that remains after one-point quantities equilibrate again vary significantly from the 0-Markov case, depending on the transition probabilities of the process. More broadly, these results identify finite temporal memory as a control handle distinct from a circuit's degree of quantum chaos. For gate sets that fall short of volume-law scrambling, the equilibrium reached by one-point diagnostics is fixed by the gates alone, while the choice of k-Markov rule controls the approach to equilibrium and the structure of persistent, late-time correlations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (25)

  1. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  2. V. Khemani, A. Vishwanath, and D. A. Huse, Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws, Phys. Rev. X 8, 031057 (2018).
  3. Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  4. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  5. A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019).
  6. M. Knap, Entanglement production and information scrambling in a noisy spin system, Phys. Rev. B 98, 184416 (2018).
  7. T. Schuster and N. Y. Yao, Operator growth in open quantum systems, Phys. Rev. Lett. 131, 160402 (2023).
  8. Z. Weinstein, S. P. Kelly, J. Marino, and E. Altman, Scrambling transition in a radiative random unitary circuit, Phys. Rev. Lett. 131, 220404 (2023).
  9. R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).
  10. J. Zhang and S. L. Braunstein, Continuous-variable Gaussian analog of cluster states, Phys. Rev. A 73, 032318 (2006).
  11. H. Buhrman, M. Folkertsma, B. Loff, and N. M. P. Neumann, State preparation by shallow circuits using feed forward, Quantum 8, 1552 (2024).
  12. L. Lovász and P. Winkler, A note on the last new vertex visited by a random walk, J. Graph Theory 17, 593 (1993).
  13. T. Gilbert and D. P. Sanders, Diffusion coefficients for multi-step persistent random walks on lattices, J. Phys. A: Math. Theor. 43, 035001 (2010).
  14. N. Balakrishnan and M. V. Koutras, Runs and Scans with Applications, Wiley Series in Probability and Statistics (John Wiley & Sons, New York, 2002).
  15. Z. Tan and P. W. Brouwer, Operator spreading in random unitary circuits with unitary-invariant gate distributions, Phys. Rev. B 111, 184301 (2025).
  16. R. Jozsa and A. Miyake, Matchgates and classical simulation of quantum circuits, Proc. R. Soc. A 464, 3089 (2008).
  17. T. Prosen and I. Pižorn, Operator space entanglement entropy in a transverse Ising chain, Phys. Rev. A 76, 032316 (2007).
  18. P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, J. High Energy Phys. 02 (2016) 004.
  19. S. Nandy, A. Sen, and D. Sen, Aperiodically driven integrable systems and their emergent steady states, Phys. Rev. X 7, 031034 (2017).
  20. S. Pilatowsky-Cameo, C. B. Dag, W. W. Ho, and S. Choi, Complete Hilbert-space ergodicity in quantum dynamics of generalized Fibonacci drives, Phys. Rev. Lett. 131, 250401 (2023).
  21. S. Pilatowsky-Cameo, S. Choi, and W. W. Ho, Critically slow Hilbert-space ergodicity in quantum morphic drives, Phys. Rev. Lett. 135, 140402 (2025).
  22. M. Bukov, L. D'Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering, Adv. Phys. 64, 139 (2015).
  23. 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).
  24. U. Akhouri, S. Shandera, and J. Henry, Open-systems tools for nonthermalizing closed quantum systems, Phys. Rev. E 112, 034113 (2025).
  25. https://github.com/seshandera/kmarkov-circuit-data.

Outline

Information

Phys. Rev. Research 8, 043016– Published 5 October, 2026
  • Received 23 March 2026
  • Accepted 31 August 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