Search

Random many-body states are both a useful tool to model certain physical systems and an important asset for quantum computation. Realising them, however, generally requires an exponential (in system size) amount of resources. Recent research has presented a way out by showing that one can generate random states, or more precisely a controlled approximation of them, by applying a quantum circuit built in terms of few-body unitary gates. Most of this research, however, has been focussed on the case of quantum circuits composed by completely random unitary gates. Here we consider what happens for circuits that, instead, involve a minimal degree of randomness. Specifically, we concentrate on two different settings: (a) brickwork quantum circuits with a single one-qudit random matrix at a boundary; (b) brickwork quantum circuits with fixed interactions but random one-qudit gates everywhere. We show that, for any given initial state, (a) and (b) produce a distribution of states approaching the Haar distribution in the limit of large circuit depth. More precisely, we show that the moments of the distribution produced by our circuits can approximate the ones of the Haar distribution in a depth proportional to the system size. Interestingly we find that in both Cases (a) and (b) the relaxation to the Haar distribution occurs in two steps - this is in contrast with what happens in fully random circuits. Moreover, we show that choosing appropriately the fixed interactions, for example taking the local gate to be a dual-unitary gate with high enough entangling power, minimally random circuits produce a Haar random distribution more rapidly than fully random circuits. In particular, dual-unitary circuits with maximal entangling power - i.e. perfect tensors - appear to provide the optimal quantum state design preparation for any design number., Comment: 17 pages, 11 figures

We obtain the exact full counting statistics of a cellular automaton with freely propagating vacancies and charged particles that are stochastically scattered or transmitted upon collision by identifying the problem as a colored stochastic six-vertex model with one inert color. Typical charge current fluctuation at vanishing net charge follow a one-parameter distribution that interpolates between the distribution of the charged single-file class in the limit of pure reflection and a Gaussian distribution in the limit of pure transmission., Comment: 35 pages, comments welcome

Symmetry resolved entanglement and entanglement asymmetry are two measures of quantum correlations sensitive to symmetries of the system. Here we discuss their non-equilibrium dynamics in the Rule 54 cellular automaton, a simple, yet interacting, integrable model. Both quantities can be expressed in terms of the more analytically tractable "charged moments", i.e. traces of powers of a suitably deformed density matrix, via a replica trick. We express them in terms of a tensor network, which we contract in space using a system of local algebraic relations. This gives the asymptotic form for the charged moments, valid in the regime of large but finite time that is shorter than all the relevant subsystem sizes. In this regime the charge moments decay exponentially with the rate given by the leading solution to a cubic equation., Comment: 21 pages, 2 figures; Submitted to the special issue of Journal of Physics A: Mathematical and Theoretical, "Quantum-Circuit Models for Many-Body Physics Out of Equilibrium"; v2 accepted version

Understanding thermalisation 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 thermalisation 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 which are only constrained to have local interactions, i.e. local random circuits. We introduce a class of random permutation circuits (RPCs), where the gates locally permute basis states modelling generic microscopic classical dynamics, and compare them to random unitary circuits (RUCs), a standard toy model for generic quantum dynamics. We show that, like RUCs, RPCs permit the analytical computation of several key quantities such as out-of-time order correlators (OTOCs), or entanglement entropies. RPCs can be interpreted both as quantum or classical dynamics, which we use to find similarities and differences between the two. Performing the average over all random circuits, we discover a series of exact relations, connecting quantities in RUC and (quantum) RPCs. In the classical setting, we obtain similar exact results relating (quantum) purity to (classical) growth of mutual information and (quantum) OTOCs to (classical) decorrelators. Our results indicate that despite of the fundamental differences between quantum and classical systems, their dynamics exhibits qualitatively similar behaviours., Comment: 26 (15+11) pages, 2 figures

We introduce a stochastic generalisation of the classical deterministic Floquet-East model, a discrete circuit with the same kinetic constraint as the East model of glasses. We prove exactly that, in the limit of long time and large size, this model has a large deviation phase transition between active and inactive dynamical phases. We also compute the finite time and size scaling of general space-time fluctuations, which for the case of inactive regions gives rise to dynamical hydrophobicity. We also discuss how, through the Trotter limit, these exact results also hold for the continuous-time East model, thus proving long-standing observations in kinetically constrained models. Our results here illustrate the applicability of exact tensor network methods for solving problems in many-body stochastic systems., Comment: 14 pages, 3 figures

The finite parts of a large, locally interacting many-body system prepared out-of-equilibrium eventually equilibrate. Characterising the underlying mechanisms of this process and its timescales, however, is particularly hard as it requires to decouple universal features from observable-specific ones. Recently, new insight came by studying how certain symmetries of the dynamics that are broken by the initial state are restored at the level of the reduced state of a given subsystem. This provides a high level, observable-independent probe. Until now this idea has been applied to the restoration of internal symmetries, e.g. U(1) symmetries related to charge conservation. Here we show that that the same logic can be applied to the restoration of space-time symmetries, and hence can be used to characterise the relaxation of fully generic systems. We illustrate this idea by considering the paradigmatic example of "generic" many-body dynamics, i.e. a local random unitary circuit. We show that, surprisingly, the restoration of translation symmetry in these systems only happens on time-scales proportional to the subsystem's volume. In fact, for large enough subsystems the time of symmetry restoration becomes initial-state independent (as long as the latter breaks the symmetry at time zero) and coincides with the thermalisation time. For intermediate subsystems, however, one can observe the so-called "quantum Mpemba effect", where the state of the system restores a symmetry faster if it is initially more asymmetric., Comment: 6+4 pages, 3 figures

We study the nonequilibrium dynamics of the Floquet quantum East model (a Trotterized version of the kinetically constrained quantum East spin chain) at its "deterministic point", where evolution is defined in terms of CNOT permutation gates. We solve exactly the thermalization dynamics for a broad class of initial product states by means of "space evolution". We prove: (i) the entanglement of a block of spins grows at most at one-half the maximal speed allowed by locality (i.e., half the speed of dual-unitary circuits); (ii) if the block of spins is initially prepared in a classical configuration, speed of entanglement is a quarter of the maximum; (iii) thermalization to the infinite temperature state is reached exactly in a time that scales with the size of the block., Comment: 19 pages, 3 figures

The highly complicated nature of far from equilibrium systems can lead to a complete breakdown of the physical intuition developed in equilibrium. A famous example of this is the Mpemba effect, which states that non-equilibrium states may relax faster when they are further from equilibrium or, put another way, hot water can freeze faster than warm water. Despite possessing a storied history, the precise criteria and mechanisms underpinning this phenomenon are still not known. Here we study a quantum version of the Mpemba effect that takes place in closed many body systems with a U(1) conserved charge: in certain cases a more asymmetric initial configuration relaxes and restores the symmetry faster than a more symmetric one. In contrast to the classical case, we establish the criteria for this to occur in arbitrary integrable quantum systems using the recently introduced entanglement asymmetry. We describe the quantum Mpemba effect in such systems and relate properties of the initial state, specifically its charge fluctuations, to the criteria for its occurrence. These criteria are expounded using exact analytic and numerical techniques in several examples, a free fermion model, the Rule 54 cellular automaton, and the Lieb-Liniger model., Comment: 6+8 pages, 2+2 figures; v2 as appears in Phys. Rev. Lett

Conserved-charge densities are very special observables in quantum many-body systems as, by construction, they encode information about the dynamics. Therefore, their evolution is expected to be of much simpler interpretation than that of generic observables and to return universal information on the state of the system at any given time. Here we study the dynamics of the fluctuations of conserved U(1) charges in systems that are prepared in charge-asymmetric initial states. We characterise the charge fluctuations in a given subsystem using the full-counting statistics of the truncated charge and the quantum entanglement between the subsystem and the rest resolved to the symmetry sectors of the charge. We show that, even though the initial states considered are homogeneous in space, the charge fluctuations generate an effective inhomogeneity due to the charge-asymmetric nature of the initial states. We use this observation to map the problem into that of charge fluctuations on inhomogeneous, charge-symmetric states and treat it using a recently developed space-time duality approach. Specialising the treatment to interacting integrable systems we combine the space-time duality approach with generalised hydrodynamics to find explicit predictions., Comment: 30 pages, 8 figures; v2 presentation significantly improved; v3 as appears in Phys. Rev. B

We study the dynamics of a classical circuit corresponding to a discrete-time kinetically constrained East model. We show that this classical "Floquet-East" model displays pre-transition behaviour which is a dynamical equivalent of the hydrophobic effect in water. For the deterministic version of the model we prove exactly: (i) a change in scaling with size in the probability of inactive space-time regions (akin to the "energy-entropy" crossover of the solvation free energy in water), (ii) a first-order phase transition in the dynamical large deviations, (iii) the existence of the optimal geometry for local phase separation to accommodate space-time solutes, and (iv) a dynamical analog of "hydrophobic collapse"., Comment: 15 pages, 5 figures; v4 final version, as appears in Phys. Rev. E

Due to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution - or its Fourier transform known as full counting statistics (FCS) - contains much more information than say the mean value of the measured observable and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables - the charged moments - that characterise how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a non-trivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out-of-equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain., Comment: 7+12 pages, 3+5 figures; v2 references added; v3 presentation improved, as appears in PRL

Strong zero modes (SZMs) are conserved operators localised at the edges of certain quantum spin chains, which give rise to long coherence times of edge spins. Here we define and analyse analogous operators in one-dimensional classical stochastic systems. For concreteness, we focus on chains with single occupancy and nearest-neighbour transitions, in particular particle hopping and pair creation and annihilation. For integrable choices of parameters we find the exact form of the SZM operators. Being in general non-diagonal in the classical basis, the dynamical consequences of stochastic SZMs are very different from those of their quantum counterparts. We show that the presence of a stochastic SZM is manifested through a class of exact relations between time-correlation functions, absent in the same system with periodic boundaries., Comment: 5+8 pages, 4 figures; v2 published version

R\'enyi entropies are conceptually valuable and experimentally relevant generalisations of the celebrated von Neumann entanglement entropy. After a quantum quench in a clean quantum many-body system they generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out-of-equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system's dynamics, and its characterisation is a key objective of current research. Here we show that the slope of R\'enyi entropies can be determined by means of a spacetime duality transformation. In essence, we argue that the slope coincides with the stationary density of entropy of the model obtained by exchanging the roles of space and time. Therefore, very surprisingly, the slope of the entanglement is expressed as an equilibrium quantity. We use this observation to find an explicit exact formula for the slope of R\'enyi entropies in all integrable models treatable by thermodynamic Bethe ansatz and evolving from integrable initial states. Interestingly, this formula can be understood in terms of a quasiparticle picture only in the von Neumann limit., Comment: 19 pages, 6 figures (v3 final version - small typos corrected)

We study the growth of entanglement between two adjacent regions in a tripartite, one-dimensional many-body system after a quantum quench. Combining a replica trick with a space-time duality transformation, we derive an exact, universal relation between the entanglement negativity and Renyi-1/2 mutual information which holds at times shorter than the sizes of all subsystems. Our proof is directly applicable to any translationally invariant local quantum circuit, i.e., any lattice system in discrete time characterised by local interactions, irrespective of the nature of its dynamics. Our derivation indicates that such a relation can be directly extended to any system where information spreads with a finite maximal velocity., Comment: 7+4 pages, 3+1 figures; v2 as appears in Phys. Rev. Lett

We introduce a pair of time-reversible models defined on the discrete space-time lattice with 3 states per site, specifically, a vacancy and a particle of two flavours (species). The local update rules reproduce the rule 54 reversible cellular automaton when only a single species of particles is present, and satisfy the requirements of flavour exchange (C), space-reversal (P), and time-reversal (T) symmetries. We find closed-form expressions for three local conserved charges and provide an explicit matrix product form of the grand canonical Gibbs states, which are identical for both models. For one of the models this family of Gibbs states seems to be a complete characterisation of equilibrium (i.e. space and time translation invariant) states, while for the other model we empirically find a sequence of local conserved charges, one for each support size larger than 2, hinting to its algebraic integrability. Finally, we numerically investigate the behaviour of spatio-temporal correlation functions of charge densities, and test the hydrodynamic prediction for the model with exactly three local charges. Surprisingly, the numerically observed 'sound velocity' does not match the hydrodynamic value. The deviations are either significant, or they decay extremely slowly with the simulation time, which leaves us with an open question for the mechanism of such a glassy behaviour in a deterministic locally interacting system., Comment: 27 pages, v2: Minor modifications, To appear in the J. Phys. A special issue "Hydrodynamics of Low-Dimensional Quantum Systems"

We study the entanglement dynamics generated by quantum quenches in the quantum cellular automaton Rule $54$. We consider the evolution from a recently introduced class of solvable initial states. States in this class relax (locally) to a one-parameter family of Gibbs states and the thermalisation dynamics of local observables can be characterised exactly by means of an evolution in space. Here we show that the latter approach also gives access to the entanglement dynamics and derive exact formulas describing the asymptotic linear growth of all R\'enyi entropies in the thermodynamic limit and their eventual saturation for finite subsystems. While in the case of von Neumann entropy we recover exactly the predictions of the quasiparticle picture, we find no physically meaningful quasiparticle description for other R\'enyi entropies. Our results apply to both homogeneous and inhomogeneous quenches., Comment: 33 pages, 5 figures; v2 accepted version

We study the out-of-equilibrium dynamics of the quantum cellular automaton Rule 54 using a time-channel approach. We exhibit a family of (non-equilibrium) product states for which we are able to describe exactly the full relaxation dynamics. We use this to prove that finite subsystems relax to a one-parameter family of Gibbs states. We also consider inhomogeneous quenches. Specifically, we show that when the two halves of the system are prepared in two different solvable states, finite subsystems at finite distance from the centre eventually relax to the non-equilibrium steady state (NESS) predicted by generalised hydrodynamics. To the best of our knowledge, this is the first exact description of the relaxation to a NESS in an interacting system and, therefore, the first independent confirmation of generalised hydrodynamics for an inhomogeneous quench., Comment: 47 pages, 3 figures; v2 accepted version

We review recent results on an exactly solvable model of nonequilibrium statistical mechanics, specifically the classical Rule 54 reversible cellular automaton and some of its quantum extensions. We discuss the exact microscopic description of nonequilibrium dynamics as well as the equilibrium and nonequilibrium stationary states. This allows us to obtain a rigorous handle on the corresponding emergent hydrodynamic description, which is treated as well. Specifically, we focus on two different paradigms of Rule 54 dynamics. Firstly, we consider a finite chain driven by stochastic boundaries, where we provide exact matrix product descriptions of the nonequilibrium steady state, most relevant decay modes, as well as the eigenvector of the tilted Markov chain yielding exact large deviations for a broad class of local and extensive observables. Secondly, we treat the explicit dynamics of macro-states on an infinite lattice and discuss exact closed form results for dynamical structure factor, multi-time-correlation functions and inhomogeneous quenches. Remarkably, these results prove that the model, despite its simplicity, behaves like a regular fluid with coexistence of ballistic (sound) and diffusive (heat) transport. Finally, we briefly discuss quantum interpretation of Rule 54 dynamics and explicit results on dynamical spreading of local operators and operator entanglement., Comment: Review paper, 70 pages, 17 figures. To appear in JSTAT special issue 2021 "Emergent Hydrodynamics in Integrable Many-body Systems"; v2 minor modifications to improve presentation

We study the out-of-equilibrium dynamics of the quantum cellular automaton known as "Rule 54". For a class of low-entangled initial states, we provide an analytic description of the effect of the global evolution on finite subsystems in terms of simple quantum channels, which gives access to the full thermalization dynamics at the microscopic level. As an example, we provide analytic formulae for the evolution of local observables and R\'enyi entropies. We show that, in contrast to other known examples of exactly solvable quantum circuits, Rule 54 does not behave as a simple Markovian bath on its own parts, and displays typical non-equilibrium features of interacting integrable many-body quantum systems such as finite relaxation rate and interaction-induced dressing effects. Our study provides a rare example where the full thermalization dynamics can be solved exactly at the microscopic level., Comment: 9 (6+3) pages; v2 as appears in Phys. Rev. Lett

We study the dynamics of a bulk deterministic Floquet model, the Rule 201 synchronous one-dimensional reversible cellular automaton (RCA201). The system corresponds to a deterministic, reversible, and discrete version of the PXP model, whereby a site flips only if both its nearest neighbours are unexcited. We show that the RCA201/Floquet-PXP model exhibits ballistic propagation of interacting quasiparticles - or solitons - corresponding to the domain walls between non-trivial three-fold vacuum states. Starting from the quasiparticle picture, we find the exact matrix product state form of the non-equilibrium stationary state for a range of boundary conditions, including both periodic and stochastic. We discuss further implications of the integrability of the model., Comment: 15 pages, 8 figures

In this paper we study the space evolution in the Rule 54 reversible cellular automaton, which is a paradigmatic example of a deterministic interacting lattice gas. We show that the spatial translation of time configurations of the automaton is given in terms of local deterministic maps with the support that is small but bigger than that of the time evolution. The model is thus an example of space-time dual reversible cellular automaton, i.e. its dual is also (in general different) reversible cellular automaton. We provide two equivalent interpretations of the result; the first one relies on the dynamics of quasi-particles and follows from an exhaustive check of all the relevant time configurations, while the second one relies on purely algebraic considerations based on the circuit representation of the dynamics. Additionally, we use the properties of the local space evolution maps to provide an alternative derivation of the matrix product representation of multi-time correlation functions of local observables positioned at the same spatial coordinate., Comment: 28 pages, 3 figures

For an interacting spatio-temporal lattice system we introduce a formal way of expressing multi-time correlation functions of local observables located at the same spatial point with a time state, i.e. a statistical distribution of configurations observed along a time lattice. Such a time state is defined with respect to a particular equilibrium state that is invariant under space and time translations. The concept is developed within the Rule 54 reversible cellular automaton, for which we explicitly construct a matrix product form of the time state, with matrices that act on the 3-dimensional auxiliary space. We use the matrix-product state to express equal-space time-dependent density-density correlation function, which, for special maximum-entropy values of equilibrium parameters, agrees with the previous results. Additionally, we obtain an explicit expression for the probabilities of observing all multi-time configurations, which enables us to study distributions of times between consecutive excitations and prove the absence of decoupling of timescales in the Rule 54 model., Comment: 17 pages, 3 figures

We discuss a simple deterministic lattice gas of locally interacting charged particles, for which we show coexistence of ballistic and diffusive transport. Both, the ballistic and the diffusive transport coefficients, specifically the Drude weight and the diffusion constant, respectively, are analytically computed for particular set of generalised Gibbs states and may independently vanish for appropriate values of thermodynamic parameters. Moreover, our analysis, based on explicit construction of the matrix representation of time-automorphism in a suitable basis of the algebra of local observables, allows for an exact computation of the dynamic structure factor and closed form solution of the inhomogeneous quench problem., Comment: 26 pages

We present an explicit time-dependent matrix product ansatz (tMPA) which describes the time-evolution of any local observable in an interacting and deterministic lattice gas, specifically for the rule 54 reversible cellular automaton of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)]. Our construction is based on an explicit solution of real-space real-time inverse scattering problem. We consider two applications of this tMPA. Firstly, we provide the first exact and explicit computation of the dynamic structure factor in an interacting deterministic model, and secondly, we solve the extremal case of the inhomogeneous quench problem, where a semi-infinite lattice in the maximum entropy state is joined with an empty semi-infinite lattice. Both of these exact results rigorously demonstrate a coexistence of ballistic and diffusive transport behaviour in the model, as expected for normal fluids., Comment: 39 pages

We developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systems where the kick strength is the small parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the un-driven (non-perturbed) Hamiltonian, while considering terms up to a finite order in the kick strength. As an application of the replica expansion, we analyze an Ising spin 1/2 chain periodically kicked with magnetic field of strength $h$, which has both longitudinal and transverse components. We demonstrate that even away from the regime of high frequency driving, the heating rate is nonperturbative in the kick strength bounded from above by a stretched exponential: $e^{-{\rm const}\,h^{-1/2}}$. This guarantees existence of a very long pre-thermal regime, where the dynamics is governed by the Floquet Hamiltonian obtained from the replica expansion., Comment: 5+4 pages, 1+3 figures

We study reversible deterministic dynamics of classical charged particles on a lattice with hard-core interaction. It is rigorously shown that the system exhibits three types of transport phenomena, ranging from ballistic, through diffusive to insulating. By obtaining an exact expressions for the current time-autocorrelation function we are able to calculate the linear response transport coefficients, such as the diffusion constant and the Drude weight. Additionally, we calculate the long- time charge profile after an inhomogeneous quench and obtain diffusive profile with the Green-Kubo diffusion constant. Exact analytical results are corroborated by Monte-Carlo simulations., Comment: 5+3 pages, 4 figures