We introduce quantum circuits in two and three spatial dimensions which are classically simulable, despite producing a high degree of operator entanglement. We provide a partial characterization of these ``automaton'' quantum circuits and use them to study operator growth, information spreading, and local charge relaxation in quantum dynamics with subsystem symmetries, which we define as overlapping symmetries that act on lower-dimensional submanifolds. With these symmetries, we discover the anomalous subdiffusion of conserved charges; that is, the charges spread slower than diffusion in the dimension of the subsystem symmetry. By studying an effective operator hydrodynamics in the presence of these symmetries, we predict the charge autocorrelator to decay (i) as $ln(t)/\sqrt{t}$ in two dimensions with a conserved U(1) charge along intersecting lines and (ii) as $1/{t}^{3/4}$ in three spatial dimensions with intersecting planar U(1) symmetries. Through large-scale studies of automaton dynamics with these symmetries, we numerically observe charge relaxation that is consistent with these predictions. In both cases, the spatial charge distribution is distinctly non-Gaussian and reminiscent of the diffusion of charges along a fractal surface. We numerically study the onset of quantum chaos in the spreading of local operators under these automaton dynamics and observe power-law broadening of the ballistically propagating fronts of evolving operators in two and three dimensions and the saturation of out-of-time-ordered correlations to values consistent with quantum chaotic behavior.
We introduce quantum circuits in two and three spatial dimensions which are classically simulable, despite producing a high degree of operator entanglement. We provide a partial characterization of these ``automaton'' quantum circuits and use them to study operator growth, information spreading, and local charge relaxation in quantum dynamics with subsystem symmetries, which we define as overlapping symmetries that act on lower-dimensional submanifolds. With these symmetries, we discover the anomalous subdiffusion of conserved charges; that is, the charges spread slower than diffusion in the dimension of the subsystem symmetry. By studying an effective operator hydrodynamics in the presence of these symmetries, we predict the charge autocorrelator to decay (i) as ln(t)/t in two dimensions with a conserved U(1) charge along intersecting lines and (ii) as 1/t3/4 in three spatial dimensions with intersecting planar U(1) symmetries. Through large-scale studies of automaton dynamics with these symmetries, we numerically observe charge relaxation that is consistent with these predictions. In both cases, the spatial charge distribution is distinctly non-Gaussian and reminiscent of the diffusion of charges along a fractal surface. We numerically study the onset of quantum chaos in the spreading of local operators under these automaton dynamics and observe power-law broadening of the ballistically propagating fronts of evolving operators in two and three dimensions and the saturation of out-of-time-ordered correlations to values consistent with quantum chaotic behavior.
The scrambling of quantum information in closed many-body systems, as measured by out-of-timeordered correlation functions (OTOCs), has received considerable attention lately.Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits.Numerical work suggests that aspects of this description are universal to ergodic many-body systems, even without randomness; a conjectured explanation for this is that while the random circuits have noise built into them, deterministic quantum systems, much like classically chaotic ones, "generate their own noise" and look effectively random on sufficient length scales and timescales.In this paper, we extend this approach to systems with locally conserved quantities (e.g., energy).We do this by considering local random unitary circuits with a conserved U(1) charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time-ordered and out-of-time-ordered correlation functions; both can have a diffusively relaxing component or "hydrodynamic tail" at late times.We verify the presence of such tails also in a deterministic, periodically driven system.We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific asymmetric shape, decaying as a power law behind the front.These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation.Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential μ, and apply perturbative arguments to show that for μ ≫ 1 the ballistic front of information spreading can only develop at times exponentially large in μ-with the information traveling diffusively at earlier times.We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.
Subdiffusion is a generic feature of chaotic many-body dynamics with multipole conservation laws and subsystem symmetries. We numerically study this subdiffusive dynamics, using quantum automaton random unitary circuits, in a broad range of models including one-dimensional models with dipole and quadrupole conservation, two-dimensional models with dipole conservation, and two-dimensional models with subsystem symmetry on the triangular lattice. Our results are in complete agreement with recent hydrodynamic predictions for such theories.
We study how conservation laws shape the spreading of quantum coherence in many-body dynamics. Focusing on U(1)-symmetric random circuits, charge-and-dipole conserving circuits, as well as ergodic Hamiltonian dynamics, we probe coherences both globally, via the participation entropy, and locally, via the relative entropy of coherence. Combining exact vector evolution, matrix product state simulations, and replica tensor networks methods, we find that conservation laws replace the logarithmic saturation of unconstrained circuits with slow hydrodynamic relaxation of the global coherence measures. Locally, symmetry-constrained circuits show a clean rise-peak-fall structure whose peak time grows algebraically with subsystem size. In contrast, ergodic Hamiltonians broaden the peak into an extended plateau at larger subsystems, highlighting a qualitatively distinct mechanism. Coherence thus emerges as a sensitive probe of symmetry-constrained thermalization, linking quantum resource dynamics to many-body transport.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
Random permutation circuits were recently introduced as minimal models for local many-body dynamics that can be interpreted both as classical and quantum. Standard dynamical complexity indicators such as damage spreading and out-of-time-order correlators (OTOCs), show that these systems exhibit sensitivity to initial conditions in the classical setting and operator scrambling in the quantum setting. Here, we address their quantum chaoticity—a stricter property—by studying the time evolution of local operator entanglement (LOE). We show that the behavior of LOE in random permutation circuits depends on the dimension of the local configuration space q . When q = 2 , i.e., the circuits act on qubits, random permutations are Clifford and the LOE of any local operator is bounded by a constant, indicating that they are not truly chaotic. On the other hand, when the dimension of the local configuration space exceeds two, the LOE grows linearly in time. We prove this in the limit of large q and present numerical evidence that a three-dimensional local configuration space is sufficient for a linear growth of LOE. Our findings highlight that quantum chaos can be produced by essentially classical dynamics. Moreover, we show that LOE can be defined also in the classical realm and put it forward as a universal indicator chaos, both quantum and classical.
Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth.We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries.We study both 1 þ 1D and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems.In 1 þ 1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed v B .We find that in 1 þ 1D, the "front" of the OTOC broadens diffusively, with a width scaling in time as t 1=2 .We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization.Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem.This implies that the width of the front of the averaged OTOC scales as t 1=3 in 2 þ 1D and as t 0.240 in 3 þ 1D (exponents of the Kardar-Parisi-Zhang universality class).We support our analytic argument with simulations in 2 þ 1D.We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical.However, when full spatial rotational symmetry is present in 2 þ 1D, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution.For an alternative perspective on the OTOC in 1 þ 1D, we map it to the partition function of an Ising-like statistical mechanics model.As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly.We also use this mapping to give exact results for entanglement growth in 1 þ 1D circuits.