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  • Open Access

Operator spreading in random unitary circuits with orthogonal/symplectic-invariant gate distributions

Zhiyang Tan (谭志阳) and Piet W. Brouwer

  • Dahlem Center for Complex Quantum Systems, Physics Department, Halle-Berlin-Regensburg Cluster of Excellence CCE, and Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

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Phys. Rev. B 114, 074308 – Published 18 August, 2026

DOI: https://doi.org/10.1103/p5d4-dmj6

Abstract

We investigate operator spreading in random quantum circuits with gates drawn from orthogonal- or symplectic-invariant ensembles, revealing several key distinctions from the well-studied unitary-invariant case. We find that the ensemble-averaged Pauli-string weights relax to a ternary-valued structure, instead of the binary structure of unitary-invariant circuits. For orthogonal- or symplectic-invariant circuits, the domain wall separating trivial and scrambled regions has a finite width even for Haar-random gates, whereas domain walls are sharp for Haar-distributed random unitary circuits. We further find a fundamental dichotomy between random circuits with two-qubit gates from the two disconnected components of the orthogonal group: While the butterfly velocity for the special orthogonal ensemble lies between zero and the Haar value, the negative-determinant sector exhibits a nonzero lower bound for any gate distribution. Moreover, for qudit size q=2, the butterfly velocity can exceed that of the Haar-random ensemble.

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References (61)

  1. L. Arnaud and D. Braun, Distribution of interference in random quantum algorithms, Phys. Rev. A 75, 062314 (2007).
  2. L. Arnaud and D. Braun, Efficiency of producing random unitary matrices with quantum circuits, Phys. Rev. A 78, 062329 (2008).
  3. W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644.
  4. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  5. 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).
  6. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  7. 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).
  8. T. Rakovszky, F. Pollmann, and C. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev. X 8, 031058 (2018).
  9. 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).
  10. M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023).
  11. B. Skinner, Lecture notes: Introduction to random unitary circuits and the measurement-induced entanglement phase transition, arXiv:2307.02986.
  12. S. Xu and B. Swingle, Locality, quantum fluctuations, and scrambling, Phys. Rev. X 9, 031048 (2019).
  13. G. Styliaris, N. Anand, and P. Zanardi, Information scrambling over bipartitions: Equilibration, entropy production, and typicality, Phys. Rev. Lett. 126, 030601 (2021).
  14. S. Xu and B. Swingle, Scrambling dynamics and out-of-time-ordered correlators in quantum many-body systems, PRX Quantum 5, 010201 (2024).
  15. X. Mi, P. Roushan, C. Quintana, S. Mandrá, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush, et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
  16. S. Vardhan and J. Wang, Free mutual information and higher-point otocs, arXiv:2509.13406.
  17. G. Q. AI, Observation of constructive interference at the edge of quantum ergodicity, Nature (London) 646, 825 (2025).
  18. Z. Tan and P. W. Brouwer, Operator spreading in random unitary circuits with unitary-invariant gate distributions, Phys. Rev. B 111, 184301 (2025).
  19. J. Haferkamp and N. Hunter-Jones, Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions, Phys. Rev. A 104, 022417 (2021).
  20. O. Shaya, Z. Holmes, C. Hirche, and A. Angrisani, On the complexity of quantum states and circuits from the orthogonal and symplectic groups, J. Phys. A: Math. Theor. 59, 205302 (2026).
  21. P. J. Forrester, Log-gases and Random Matrices (Princeton University Press, Princeton, NJ, 2010).
  22. M. L. Mehta, Random Matrices (Elsevier, Amsterdam, 2004).
  23. A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman, Real randomized benchmarking, Quantum 2, 85 (2018).
  24. M. West, A. A. Mele, M. Larocca, and M. Cerezo, Random ensembles of symplectic and unitary states are indistinguishable, arXiv:2409.16500.
  25. T. Schuster, J. Haferkamp, and H.-Y. Huang, Random unitaries in extremely low depth, Science 389, 92 (2025).
  26. L. Grevink, J. Haferkamp, M. Heinrich, J. Helsen, M. Hinsche, T. Schuster, and Z. Zimborás, Will it glue? On short-depth designs beyond the unitary group, arXiv:2506.23925.
  27. A. Sauliere, B. Magni, G. Lami, X. Turkeshi, and J. De Nardis, Universality in the anticoncentration of chaotic quantum circuits, Phys. Rev. B 112, 134312 (2025).
  28. B. Magni, M. Heinrich, L. Leone, and X. Turkeshi, Anticoncentration and state design of doped real Clifford circuits and tensor networks, Phys. Rev. A, 062446 (2026) 113.
  29. M. West, A. Anna Mele, M. Larocca, and M. Cerezo, Real classical shadows, J. Phys. A: Math. Theor. 58, 245304 (2025).
  30. K. Khanna, A. Kumar, R. Vasseur, and A. W. W. Ludwig, Random quantum circuits with time-reversal symmetry, Phys. Rev. Res. 8, 013165 (2026).
  31. N. Hunter-Jones and J. Haferkamp, Ideal random quantum circuits pass the LXEB test, arXiv:2602.22692.
  32. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  33. A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  34. A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009).
  35. N. Hunter-Jones, Operator growth in random quantum circuits with symmetry, arXiv:1812.08219.
  36. F. J. Dyson, A Brownian-Motion Model for the Eigenvalues of a Random Matrix, J. Math. Phys. 3, 1191 (1962).
  37. F. J. Dyson, A class of matrix ensembles, J. Math. Phys. 13, 90 (1972).
  38. M. L. Mehta and A. Pandey, On some Gaussian ensembles of Hermitian matrices, J. Phys. A: Math. Gen. 16, 2655 (1983).
  39. M. L. Mehta, Random matrices and matrix models: The JNU lectures, Pramana - J. Phys. 48, 7 (1997).
  40. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge university press, Cambridge, 2012).
  41. B. Bertini, P. W. Claeys, and T. Prosen, Exactly solvable quantum many-body dynamics from space-time duality, Rev. Mod. Phys. 98, 025001 (2026).
  42. P. Zanardi, C. Zalka, and L. Faoro, Entangling power of quantum evolutions, Phys. Rev. A 62, 030301(R) (2000).
  43. X. Wang, B. C. Sanders, and D. W. Berry, Entangling power and operator entanglement in qudit systems, Phys. Rev. A 67, 042323 (2003).
  44. T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev. B 99, 174205 (2019).
  45. R. Suzuki, H. Katsura, Y. Mitsuhashi, T. Soejima, J. Eisert, and N. Yoshioka, More global randomness from less random local gates, arXiv:2410.24127.
  46. Y. Mitsuhashi, R. Suzuki, T. Soejima, and N. Yoshioka, Unitary designs of symmetric local random circuits, Phys. Rev. Lett. 134, 180404 (2025).
  47. J. Patera and H. Zassenhaus, The Pauli matrices in n dimensions and finest gradings of simple Lie algebras of type An−1, J. Math. Phys. 29, 665 (1988).
  48. P. W. Brouwer and C. W. J. Beenakker, Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems, J. Math. Phys. 37, 4904 (1996).
  49. D. Weingarten, Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys. 19, 999 (1978).
  50. S. Samuel, U(N) integrals, 1/N, and the De Wit-'t Hooft anomalies, J. Math. Phys. 21, 2695 (1980).
  51. B. Collins and P. Śniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Commun. Math. Phys. 264, 773 (2006).
  52. S. Matsumoto, Weingarten calculus for matrix ensembles associated with compact symmetric spaces, Random Matrices: Theory Appl. 02, 1350001 (2013).
  53. R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. Math. 38, 857 (1937).
  54. Z. Tan and P. W. Brouwer, Brownian motion in orthogonal and symplectic groups, arXiv:2607.05094.
  55. H. D. Macedo and J. N. Oliveira, Typing linear algebra: A biproduct-oriented approach, Sci. Comput. Program. 78, 2160 (2013).
  56. D. Dummit and R. Foote, Abstract Algebra (Wiley, Hoboken, NJ, 2003).
  57. J. B. Fraleigh, A First Course in Abstract Algebra (Pearson Education, Boston, MA, 2003).
  58. M. R. Sepanski, Compact Lie Groups (Springer, New York, 2007).
  59. W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics (Springer, New York, 2004), Vol. 129.
  60. B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics (Springer, New York, 2003).
  61. L. C. Grove, Classical Groups and Geometric Algebra, Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2002), Vol. 39.

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Phys. Rev. B 114, 074308– Published 18 August, 2026
  • Received 2 June 2026
  • Revised 4 August 2026
  • Accepted 5 August 2026
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