The process by which open quantum systems thermalize with an environment is both of fundamental interest and relevant to noisy quantum devices. As a minimal model of this process, we consider a qudit chain evolving under local random unitaries and local depolarization channels. After mapping to a statistical mechanics model, the depolarization (noise) acts like a symmetry-breaking field, and we argue that it causes the system to thermalize within a timescale independent of system size. We show that various bipartite entanglement measures---mutual information, operator entanglement, and entanglement negativity---grow at a speed proportional to the size of the bipartition boundary. As a result, these entanglement measures obey an area law: Their maximal value during the dynamics is bounded by the boundary instead of the volume. In contrast, if the depolarization only acts at the system boundary, then the maximum value of the entanglement measures obeys a volume law. We complement our analysis with scalable simulations involving Clifford gates, for both one- and two-dimensional systems.
The process by which open quantum systems thermalize with an environment is both of fundamental interest and relevant to noisy quantum devices. As a minimal model of this process, we consider a qudit chain evolving under local random unitaries and local depolarization channels. After mapping to a statistical mechanics model, the depolarization (noise) acts like a symmetry-breaking field, and we argue that it causes the system to thermalize within a timescale independent of system size. We show that various bipartite entanglement measures---mutual information, operator entanglement, and entanglement negativity---grow at a speed proportional to the size of the bipartition boundary. As a result, these entanglement measures obey an area law: Their maximal value during the dynamics is bounded by the boundary instead of the volume. In contrast, if the depolarization only acts at the system boundary, then the maximum value of the entanglement measures obeys a volume law. We complement our analysis with scalable simulations involving Clifford gates, for both one- and two-dimensional systems.
A random quantum circuit is a minimally structured model to study entanglement dynamics of many-body quantum systems. We consider a one-dimensional quantum circuit with noisy Haar-random unitary gates using density matrix operator and tensor contraction methods. It is shown that the entanglement evolution of the random quantum circuits is properly characterized by the logarithmic entanglement negativity. By performing exact numerical calculations, we find that, as the physical error rate is decreased below a critical value p c ≈ 0.056, the logarithmic entanglement negativity changes from the area law to the volume law, giving rise to an entanglement transition. The critical exponent of the correlation length can be determined from the finite-size scaling analysis, revealing the universal dynamic property of the noisy intermediate-scale quantum devices.
Whether noisy quantum devices without error correction can provide quantum advantage over classical computers is a critical issue of current quantum computation. In this work, the random quantum circuits, which are used as the paradigm model to demonstrate quantum advantage, are simulated with depolarizing noise on experiment-relevant two-dimensional architecture. With comprehensive numerical simulation and theoretical analysis, we find that the maximum achievable operator entanglement entropy, which indicates maximal simulation cost, has area law scaling with the system size for constant noise rate. On the other hand, we also find that the maximum achievable operator entanglement entropy has power-law scaling with the noise rate for fixed system size, and the volume law scaling can be obtained only if the noise rate decreases when system size increase.
We study measurement-induced entanglement (MIE) generated by column-by-column sampling of noisy 2D random circuits of size N and depth T . Focusing primarily on Clifford circuits and using the operator entanglement S op of the sampling-induced boundary state as a proxy for computational complexity, first, we reproduce in the noiseless limit a finite-depth transition from area- to volume-law scaling at a threshold depth T c = 6 . In contrast, in the presence of single-qubit depolarizing noise at any constant rate p > 0 , we find that the operator entanglement S op obeys an area law, with its maximum value scaling approximately linearly with T / p in the regime T > T c . By analyzing the spatial distribution of stabilizer generators, we observe exponential localization of stabilizer generators; these both account for the scaling of the maximal S op and imply an exponential decay of conditional mutual information across buffered tripartitions, which we also confirm numerically. Together, these results indicate that constant local noise destroys long-range MIE in 2D random Clifford circuits, and that a tensor-network-based algorithm can efficiently sample from noisy 2D random Clifford circuits (i) at sub-logarithmic depths T = o ( log N ) for any constant noise rate p = Ω ( 1 ) , and (ii) at constant depths T = O ( 1 ) for noise rates p = Ω ( log − 1 N ) . Finally, we turn to depth T = 4 Haar-random and measurement-based quantum computing-type circuits, providing evidence that MIE in noisy 2D Haar-random circuits exhibits the same qualitative behavior as in random Clifford circuits, and that noise destroys the volume-law scaling of MIE in non-Clifford circuits.
We study the dynamics of entanglement asymmetry in random unitary circuits (RUCs). Focusing on a local U ( 1 ) charge, we consider symmetric initial states evolved by both local one-dimensional circuits and geometrically nonlocal RUCs made of two-qudit gates. We compute the entanglement asymmetry of subsystems of arbitrary size, analyzing the relaxation timescales. We show that the entanglement asymmetry of the whole system approaches its stationary value in a time independent of the system size for both local and nonlocal circuits. For subsystems, we find qualitative differences depending on their size. When the subsystem is larger than half of the full system, the equilibration timescales are again independent of the system size for both local and nonlocal circuits and the entanglement asymmetry grows monotonically in time. Conversely, when the subsystems are smaller than half of the full system, we show that the entanglement asymmetry is nonmonotonic in time and that it equilibrates in a time proportional to the quantum-information scrambling time, providing a physical intuition. As a consequence, the subsystem-equilibration time depends on the locality of interactions, scaling linearly and logarithmically in the system size, respectively, for local and nonlocal RUCs. Our work confirms the entanglement asymmetry as a versatile and computable probe of symmetry in many-body physics and yields a phenomenological overview of entanglement-asymmetry evolution in typical nonintegrable dynamics.
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.
Generic many-body systems coupled to an environment lose their quantum entanglement due to decoherence and evolve to a mixed state with only classical correlations. Here, we show that measurements can stabilize quantum entanglement within open quantum systems. Specifically, in random unitary circuits with dephasing at the boundary, we find both numerically and analytically that projective measurements performed at a small nonvanishing rate result in a steady state with an L^{1/3} power-law scaling entanglement negativity within the system. Using an analytical mapping to a statistical mechanics model of directed polymers in a random environment, we show that the power-law negativity scaling can be understood as Kardar-Parisi-Zhang fluctuations due to the random measurement locations. Further increasing the measurement rate leads to a phase transition into an area-law negativity phase, which is of the same universality as the entanglement transition in monitored random circuits without decoherence.