Decoupling has become a central concept in quantum information theory with applications including proving coding theorems, randomness extraction and the study of conditions for reaching thermal equilibrium. However, our understanding of the dynamics that lead to decoupling is limited. In fact, the only families of transformations that are known to lead to decoupling are (approximate) unitary two-designs, i.e., measures over the unitary group which behave like the Haar measure as far as the first two moments are concerned. Such families include for example random quantum circuits with O(n 2 ) gates, where n is the number of qubits in the system under consideration. In fact, all known constructions of decoupling circuits use Ω(n 2 ) gates. Here, we prove that random quantum circuits with O(n log 2 n) gates satisfy an essentially optimal decoupling theorem. In addition, these circuits can be implemented in depth O(log 3 n). This proves that decoupling can happen in a time that scales polylogarithmically in the number of particles in the system, provided all the particles are allowed to interact. Our proof does not proceed by showing that such circuits are approximate two-designs in the usual sense, but rather we directly analyze the decoupling property.

The phenomenon of many-body localisation received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at non-zero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalisation following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties -a vanishing group velocity and the absence of transport -with entanglement properties of individual eigenvectors. Using Lieb-Robinson bounds and filter functions, we prove rigorously under simple assumptions on the spectrum that if a system shows strong dynamical localisation, all of its manybody eigenvectors have clustering correlations. In one dimension this implies directly an entanglement area law, hence the eigenvectors can be approximated by matrix-product states. We also show this statement for parts of the spectrum, allowing for the existence of a mobility edge above which transport is possible.The concept of disorder induced localisation has been introduced in the seminal work by Anderson [1] who captured the mechanism responsible for the absence of diffusion of waves in disordered media. This mechanism is specifically well understood in the single-particle case, where one can show that in the presence of a suitable random potential, all eigenfunctions are exponentially localised [1]. In addition to this spectral characterisation of localisation there is a notion of dynamical localisation, which requires that the transition amplitudes between lattice sites decay exponentially [2,3].Naturally, there is a great interest in extending these results to the many-body setting [4,5]. In the case of integrable systems that can be mapped to free fermions, such as the XY chain, results on single particle localisation can be applied directly [6,7]. A far more intricate situation arises in interacting systems. Such many-body localisation [8,9] has received an enormous attention recently. In terms of condensed-matter physics, this phenomenon allows for systems to remain an insulator even at non-zero temperature [10], in principle even at infinite temperature [11]. In the foundations of statistical mechanics, such many-body localised systems provide examples of systems that fail to thermalise. When pushed out of equilibrium, signatures of the initial condition will locally be measurable even after long times, in contradiction to what one might expect from quantum statistical mechanics [4,12,13].Despite great efforts to approach the phenomenon of manybody localisation, many aspects are not fully understood and a comprehensive definition is still lacking. Similar to the case of single-particle Anderson localisation, there are two complementary approaches to capture the phenomenon. On the one hand probes involving real-time dynamics [14,15] have been discussed, showing excitations "getting stuck", or seeing suitable signatures in density-auto-correlation functions [16], le...

We consider a class of random quantum circuits where at each step a gate from a universal set is applied to a random pair of qubits, and determine how quickly averages of arbitrary finite-degree polynomials in the matrix elements of the resulting unitary converge to Haar measure averages. This is accomplished by establishing an exact mapping between the superoperator that describes t-order moments on n qubits and a multilevel SU (4 t ) Lipkin-Meshkov-Glick Hamiltonian. For arbitrary fixed t, we find that the spectral gap scales as 1/n in the thermodynamic limit. Our results imply that random quantum circuits yield an efficient implementation of ǫ-approximate unitary t-designs. [6]. That is, a unitary t-design faithfully simulates the Haar measure with respect to any test that uses at most t copies of a selected n-qubit unitary. Ramifications of the theory of t-designs [8] are being uncovered in problems as different as black hole evaporation and fast "scrambling" of information [9], efficient quantum tomography and randomized gate benchmarking [10], quantum channel capacity [11], and the foundations of quantum statistical mechanics [12].Prompted by the above advances, significant effort has been devoted recently to identifying efficient constructions of t-designs and characterizing their convergence properties [2,6,[13][14][15][16]. Harrow and Low established, in particular, the equivalence between approximate 2-designs and random quantum circuits as introduced in [4], and conjectured that a random circuit consisting of k=poly(n, t) gates from a two-qubit universal gate set yields an approximate t-design [16]. While supporting numerical evidence was gathered in [17] for low-order moments, and efficient constructions of t-designs were reported in [16] for any t = O(n/ log n), the extent to which random quantum circuits could be used to implement an approximate t-design for arbitrary, fixed t remained open.In this Letter, we address this question by determining the rate at which, for sufficiently large circuit depth, statistical moments of arbitrary order converge to their limiting Haar values. Our strategy involves two steps: first, for given t, we show that the asymptotic convergence rate is determined by the spectral gap of a certain superoperator, which encapsulates moments up to order t; next, we compute this gap by mapping the t-moment superoperator to a multilevel version of the Lipkin-Meshkov-Glick (LMG) model, which is known to be exactly solvable, and whose low-energy spectrum is well understood in the thermodynamic limit n → ∞ [18]. Our approach ties together t-design theory with established mean-field techniques from many-body physics, extending earlier results by Znidaric [14] for t = 2. Furthermore, asymptotic convergence rates allow us to upper bound the convergence time (minimum circuit length, k c ) needed for a desired accuracy ǫ relative to the Haar measure to be reached. For any fixed t, we find that the scaling k c ∼ n log(1/ǫ) holds for sufficiently large n and small ǫ.Moment superoperator.−...

We investigate disordered one-and two-dimensional Heisenberg spin lattices across a transition from integrability to quantum chaos from both a statistical many-body and a quantum-information perspective. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implications of basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin-and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.