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Random quantum circuits with time-reversal symmetry

Kabir Khanna1,2, Abhishek Kumar2, Romain Vasseur1,*, and Andreas W. W. Ludwig3

  • *Contact author: romain.vasseur@unige.ch
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Phys. Rev. Research 8, 013165 – Published 12 February, 2026

DOI: https://doi.org/10.1103/4bgj-p2jk

Abstract

Time-reversal (TR) symmetry is crucial for understanding a wide range of physical phenomena, and plays a key role in constraining fundamental particle interactions and in classifying phases of quantum matter. In this work, we introduce an ensemble of random quantum circuits that are representative of the dynamics of generic TR-invariant many-body quantum systems. We derive a general statistical mechanics model describing entanglement, many-body quantum chaos, and quantum information dynamics in such TR-invariant circuits. As an example of application of our formalism, we study the universal properties of measurement-induced phase transitions in monitored TR-invariant systems, with measurements performed in a TR-invariant basis. We find that TR invariance of the unitary part of the dynamics does not affect the universality class, unless measurement outcomes are postselected to satisfy the global TR invariance of each quantum trajectory. We confirm these predictions numerically and find, for both generic and Clifford-based evolutions, critical exponents in the case of “strong,” i.e., global TR invariance where each quantum trajectory is TR invariant.

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  143. There is a minor but subtle change to this statement for the global postselected scenario. Using Eq. (18), we see that the replicated annealed averaged partition function Z¯k=∑mpmkqm, where we use the born probability qm≠pm of only the first half of the trajectory to weigh each term in the partition function Zmk. It follows from this that F=−∑mqmlnpm for the global postselected case, which is not the (Shannon) entropy of the measurement record as is the case for the local TR-symmetric and Haar models.

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Phys. Rev. Research 8, 013165– Published 12 February, 2026
  • Received 11 April 2025
  • Accepted 27 November 2025
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