Random Quantum Circuits with Time-Reversal Symmetry

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 a ensemble of random quantum circuits which are representative of the dynamics of generic TR-invariant many-body quantum systems. We derive a general statistical mechanics model describing entanglement and quantum information dynamics in such circuits. As an example of application of our formalism, we study the universal properties of measurement-induced phase transitions (MIPT) 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 post-selected to satisfy the global TR-invariance of each quantum trajectory. We confirm these predictions numerically, and find novel critical exponents in the case of ``strong'' TR-invariance where each quantum trajectory is TR-invariant.