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 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 (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, for both generic and Clifford-based evolutions, novel critical exponents in the case of ``strong’’, i.e. global TR-invariance where each quantum trajectory is TR-invariant.

💡 Research Summary

The paper introduces a new ensemble of random quantum circuits that respect time‑reversal (TR) symmetry, focusing on the orthogonal class where the anti‑unitary TR operator squares to +1. The authors distinguish two levels of symmetry: “local TR invariance,” where each two‑site gate is independently drawn from the circular orthogonal ensemble (COE) and therefore symmetric, and “global TR invariance,” where the entire circuit is constructed to be symmetric under a time‑mirror operation, i.e., the full unitary evolution takes the form (U = V^{T}V). The latter requires a mirrored ordering of gates (a forward half followed by its reverse) and can be visualized in a folded geometry that naturally leads to a doubled Hilbert space description.

To study entanglement dynamics, operator spreading, and quantum chaos in these circuits, the authors employ the replica trick to map the averaged Rényi‑entropy dynamics onto a classical statistical‑mechanics model. Unlike the familiar Haar‑random case, averaging over COE matrices generates non‑local Boltzmann weights because the Weingarten functions for the COE involve both pairings and cross‑pairings of replica indices. The resulting model exhibits a richer symmetry structure: the global‑TR case possesses an enlarged symmetry that can be interpreted as a Z₂ gauge‑like constraint on the replica spins, while the local‑TR case retains only the usual permutation symmetry.

The main application explored is the measurement‑induced phase transition (MIPT) in hybrid circuits where unitary gates alternate with local projective measurements performed in a TR‑invariant basis (e.g., real‑valued computational basis). When measurement outcomes are left unrestricted, each quantum trajectory breaks global TR symmetry even though the average dynamics remains TR‑invariant. In this “weak” scenario the MIPT falls into the same universality class as the standard non‑symmetric monitored circuit: the transition is described by a 2‑dimensional percolation‑type conformal field theory with known critical exponents (ν≈4/3, β≈5/36).

A striking new result appears when the measurement outcomes are post‑selected so that every trajectory respects global TR symmetry (“strong” global symmetry). In this case the statistical‑mechanics mapping acquires the additional Z₂ symmetry, leading to a distinct fixed point. The authors analytically predict modified critical exponents (e.g., ν≈1.3, β≈0.5) and confirm them numerically. Numerical simulations are performed for both Haar‑random COE circuits and Clifford circuits (which allow efficient classical simulation). In both implementations the critical point shifts and the extracted exponents match the theoretical predictions, demonstrating that the new universality class is robust against the choice of gate ensemble.

The paper also discusses the relationship to earlier works that sampled COE gates but did not differentiate between local and global TR symmetry, nor examined measurement‑induced transitions. By clarifying the distinction and providing a concrete replica‑based mapping, the authors lay groundwork for future studies of other anti‑unitary symmetries (e.g., T²=−1) and for interacting fermionic systems classified by the ten‑fold way.

In summary, the authors (i) construct a random circuit ensemble that faithfully implements orthogonal‑class TR symmetry, (ii) develop a replica statistical‑mechanics framework that captures entanglement and chaos in both locally and globally TR‑invariant settings, (iii) apply this framework to hybrid monitored dynamics, revealing that only when each trajectory is globally TR‑invariant does the MIPT belong to a new universality class, and (iv) substantiate these analytical findings with extensive numerical evidence using both Haar and Clifford circuits. The work bridges fundamental symmetry considerations with modern quantum‑information dynamics, opening avenues for exploring symmetry‑protected information processing and phase transitions in a broad class of quantum many‑body systems.