Random unitary circuits are powerful minimal models for chaotic nonequilibrium evolution. Here, the authors establish an exact mapping between dynamical observables in random circuits and a statistical mechanical model of interacting spins, and use it to study systematically the entanglement generated by a random circuit consisting of local gates. A domain wall line tension in this spin model sets the entanglement growth rate. By a careful analysis of the interaction between domain walls, the authors derive explicitly the Kardar-Parisi-Zhang equation that governs the evolution of second Renyi entropy and identify a phase transition of the growth-rate function.
Random unitary circuits are powerful minimal models for chaotic nonequilibrium evolution. Here, the authors establish an exact mapping between dynamical observables in random circuits and a statistical mechanical model of interacting spins, and use it to study systematically the entanglement generated by a random circuit consisting of local gates. A domain wall line tension in this spin model sets the entanglement growth rate. By a careful analysis of the interaction between domain walls, the authors derive explicitly the Kardar-Parisi-Zhang equation that governs the evolution of second Renyi entropy and identify a phase transition of the growth-rate function.
Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the “entanglement tsunami” in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar-Parisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like (time)[superscript 1/3] and are spatially correlated over a distance ∝(time)[superscript 2/3]. We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a “minimal cut” picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the “velocity” of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder. Subject Areas: Condensed Matter Physics, Quantum Information, Statistical Physics
Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary k-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on n qudits form approximate 2-designs in O(n) depth, as is known. Furthermore, we argue that random circuits form approximate unitary k-designs in O(nk) depth and are thus essentially optimal in both n and k. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.
In certain analytically tractable quantum chaotic systems, the calculation of out-of-time-order correlation functions, entanglement entropies after a quench, and other related dynamical observables reduces to an effective theory of an "entanglement membrane" in spacetime.These tractable systems involve an average over random local unitaries defining the dynamical evolution.We show here how to make sense of this membrane in more realistic models, which do not involve an average over random unitaries.Our approach relies on introducing effective pairing degrees of freedom in spacetime, describing a pairing of forward and backward Feynman trajectories, inspired by the structure emerging in random unitary circuits.This viewpoint provides a framework for applying ideas of coarse graining to dynamical quantities in chaotic systems.We apply the approach to some translationally invariant Floquet spin chains studied in the literature.We show that a consistent line tension may be defined for the entanglement membrane and that there are qualitative differences in this tension between generic models and "dualunitary" circuits.These results allow scaling pictures for out-of-time-order correlators and for entanglement to be taken over from random circuits to nonrandom Floquet models.We also provide an efficient numerical algorithm for determining the entanglement line tension in 1 þ 1D.
The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth Rényi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-Rényi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.
Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements.We study how entanglement dynamics in nonunitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers.We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system.We find that while Rényi entropies grow sub-ballistically as ffiffi t p in the absence of measurement, for even an infinitesimal rate of measurements, all average Rényi entropies grow ballistically with time ∼t.We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes.Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively stabilized symmetry-breaking and topological orders.
Dissipation is inevitable in realistic quantum circuits. We examine the effects of dissipation on a class of monitored random circuits that exhibit a measurement-induced entanglement phase transition. This transition has previously been understood as an order-to-disorder transition of an effective classical spin model. We extend this mapping to include on-site dissipation described by the dephasing and amplitude damping channel and study the corresponding 2D Ising model with generalized interactions and develop diagrammatic methods for the exact Boltzmann weights of the bonds in terms of probability of measurement p, the dissipation rate $\mathrm{\ensuremath{\Gamma}}$, and the on-site Hilbert space dimension q. The dissipation plays the role of Z2-symmetry-breaking interactions, while small measurement rates reduces the ratio of the symmetry-breaking interactions to the pairwise interactions, conducive to long-range order. We analyze the dynamical regimes of the R\'enyi mutual information and find that the joint action of monitored measurements and dissipation yields short time, intermediate time and steady-state behavior that can be understood in terms of crossovers between different classical domain wall configurations. The presented analysis applies to monitored open or Lindbladian quantum systems and provides a tool to understand entanglement dynamics in realistic dissipative settings and achievable system sizes.