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.
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.
Understanding thermalization in quantum many-body systems is among the most enduring problems in modern physics. A particularly interesting question concerns the role played by quantum mechanics in this process, i.e., whether thermalization in quantum many-body systems is fundamentally different from that in classical many-body systems and, if so, which of its features are genuinely quantum. Here, we study this question in minimally structured many-body systems that are only constrained to have local interactions, i.e., local random circuits. In particular, we introduce random permutation circuits (RPCs), which are circuits comprising gates that locally permute basis states, as a counterpart to random unitary circuits (RUCs), a standard toy model for generic quantum dynamics. RPCs represent a model for generic microscopic classical reversible dynamics but, interestingly, can be interpreted both as classical or as quantum dynamics. We show that, upon averaging over all circuit realizations, RPCs permit the analytical computation of several key quantities such as out-of-time-order correlators (OTOCs) and entanglement entropies. In the classical setting, we obtain similar exact results relating (quantum) purity to (classical) growth of mutual information and (quantum) OTOCs to (classical) decorrelators. We, thus, discover a series of exact relations, connecting quantities in RUC and (quantum or classical) RPCs. Our results indicate that, despite the fundamental differences between quantum and classical systems, their many-body dynamics exhibits remarkably similar behaviors.
Quantum chaos entails an entropic and computational obstruction to describing a system and thus is intrinsically difficult to characterize. An understanding of quantum chaos is fundamentally related to the mechanism of thermalization in many-body systems and the quantum nature of black holes. In this thesis we adopt the view that quantum information theory provides a powerful framework in which to elucidate chaos in strongly-interacting quantum systems. We first push towards a more precise understanding of chaotic dynamics by relating different diagnostics of chaos, studying the time-evolution of random matrix Hamiltonians, and quantifying random matrix behavior in physical systems. We derive relations between out-of-time ordered correlation functions, spectral quantities, and frame potentials to relate the scrambling of quantum information, decay of correlators, and Haar-randomness. We give analytic expressions for these quantities in random matrix theory to explore universal aspects of late-time dynamics. Motivated by our random matrix results, we define k-invariance in order to capture the onset of random matrix behavior in physical systems. We then refine our diagnostics in order to study chaotic systems with symmetry by considering Haar-randomness with respect to quotients of the unitary group, and in doing so we generalize our quantum information machinery. We further consider extended random matrix ensembles in the context of strongly-interacting quantum systems dual to black holes. Lastly, we study operator growth in classes of random quantum circuits.
We investigate universal signatures of quantum chaos in the presence of time reversal symmetry (TRS) in generic many body quantum chaotic systems (gMBQCs). We study three classes of minimal models of gMBQCs with TRS, realized through random quantum circuits with (i) local TRS, (ii) global TRS, and (iii) TRS combined with discrete time-translation symmetry. In large local Hilbert space dimension q, we derive the emergence of random matrix theory (RMT) universality in the spectral form factor (SFF) at times larger than the Thouless time tTh, which diverges with system sizes in gMBQCs. At times before tTh, we identify universal behaviour beyond RMT by deriving explicit scaling functions of SFF in the thermodynamic limit. In particular, in the simplest non-trivial setting - preserving global TRS while breaking time translation symmetry - we show that the SFF is mapped to the partition function of an emergent classical ferromagnetic Ising model, where the Ising spins correspond to the time-parallel and time-reversed pairings of Feynman paths, and external magnetic fields are induced by TRS-breaking mechanisms. Without relying on the large-q limit, we develop a second independent derivation of the Ising scaling behaviour of SFF using space-time duality and parity symmetric non-Hermitian Ginibre ensembles. Moreover, we show that many body effects originating from time-reversed pairings of Feynman paths manifest in the two-point autocorrelation function (2PAF), the out-of-time-ordered correlator (OTOC), and the partial spectral form factor - quantities sensitive to both eigenvalue and eigenstate correlations. We establish that the fluctuations of 2PAF are governed by an emergent three-state Potts model, leading to an exponential scaling with the operator support size, at a rate set by the three-state Potts model. [See full abstract in the paper]
Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution. Bipartite entanglement measures can be used to characterize the dynamical phases that emerge in these systems. Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions, with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators. In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.
We study the critical properties of random quantum circuits with a U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)-symmetric Haar and stabilizer random circuits.
We study the critical properties of random quantum circuits with a U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)-symmetric Haar and stabilizer random circuits.