We study the emergent randomness in projected ensembles generated by dual-unitary circuits. Specifically, we consider evolution by solvable dual-unitary brickwork circuits followed by measurements on part of the system, which induce an ensemble of states on the unmeasured subsystem. Previous work has established that this projected ensemble of states forms a quantum state design in the limit where the bath size is large, but the rate of convergence at finite bath size has not been rigorously established. Under three inverse-polynomial assumptions, local permutation mixing, a bound on recurrent backward motion, and contraction at the measured boundary, we fill this gap and prove that a bath of polynomial size suffices to form approximate $k$-designs. In establishing our results, we develop a microscopic transport interpretation of emergent randomness in the projected ensemble, in which deviations from local permutation operators appear as defects, and make connections with the theory of quantum walks.
We study the emergent randomness in projected ensembles generated by dual-unitary circuits. Specifically, we consider evolution by solvable dual-unitary brickwork circuits followed by measurements on part of the system, which induce an ensemble of states on the unmeasured subsystem. Previous work has established that this projected ensemble of states forms a quantum state design in the limit where the bath size is large, but the rate of convergence at finite bath size has not been rigorously established. Under three inverse-polynomial assumptions, local permutation mixing, a bound on recurrent backward motion, and contraction at the measured boundary, we fill this gap and prove that a bath of polynomial size suffices to form approximate k-designs. In establishing our results, we develop a microscopic transport interpretation of emergent randomness in the projected ensemble, in which deviations from local permutation operators appear as defects, and make connections with the theory of quantum walks.
Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble . In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design . Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.
Recent studies of "deep thermalization" have revealed universal physics in quantum many-body dynamics beyond equilibration towards Gibbs states: maximally random quantum state ensembles can emerge on local subsystems, generated by measurements on their complement. In this work, we further identify a new form of universality exhibited in the "finer fingerprints" of quantum dynamics for local subsystems, induced by global unitary time-evolution. Specifically, we consider the projected Kraus ensemble, an ensemble of Kraus operators obtained by unraveling the quantum channel on a small subsystem with respect to knowledge of the classical configurations of its complement. Our central result is a one-parameter random matrix Ansatz that captures the ensemble's emergent statistical behavior along particular spacetime scalings, valid for generic 1D circuit dynamics without conservation laws: the ensemble is described by the product of a complex Ginibre random matrix and an independent log-normal random real scalar. The former encodes scrambling within the subsystem, captured by rotational invariance of the Ginibre measure, whereas the latter encodes fluctuations in the Born probabilities, arising from locality of the underlying dynamics. Our Ansatz can be established in the special cases of dynamics generated by global Haar random unitaries and dual-unitary circuits, while we motivate it for generic circuits using arguments of spacetime duality and the multiplicative ergodic theorem on long products of spatial transfer matrices. Extensive numerical simulations for random and Floquet circuit models verify these predictions. This universality in Kraus operators also provides a microscopic mechanism for deep thermalization in generic 1D quantum circuit dynamics, and has implications for local quantum information recoverability involving classical side information tied to knowledge of the bath.
Quantum thermalization in a many-body system is defined by the approach of local subsystems toward a universal form, describable as an ensemble of quantum states wherein observables acquire thermal expectation values.Recently, it has been demonstrated that the distribution of these quantum states can also exhibit universal statistics, upon associating each state with the outcome of a local projective measurement of the complementary subsystem.Specifically, this collection of pure quantum states-called the projected ensemble-can, under certain conditions, mimic the behavior of a maximally entropic uniformly random ensemble, i.e., form a quantum state design, representing a "deeper" form of quantum thermalization.In this work, we investigate the dynamical process underlying this novel emergent universality.Leveraging a space-time duality mapping for one-dimensional quantum circuits, we argue that the physics of dynamical purification, which arises in the context of monitored quantum systems, constrains the approach of the projected ensemble toward the uniform distribution.We prove that the absence of dynamical purification in the space-time dual dynamics (a condition realized in dual-unitary quantum circuits with appropriate initial states and final measurement bases) generically yields exact state designs for all moments k at the same time, extending previous rigorous results [Ho and Choi, Phys.Rev. Lett.128, 060601 (2022)].Conversely, we show that, departing from these conditions, dynamical purification can lead to a separation of time scales between the formation of a quantum state design for moment k = 1 (regular thermalization) and for high moments k 1 (deep thermalization).Our results suggest that the projected ensemble can probe nuanced features of quantum dynamics inaccessible to regular thermalization, such as quantum information scrambling.
Abstract The projected ensemble is based on the study of the quantum state of a subsystem A conditioned on projective measurements in its complement. Recent studies have observed that a more refined measure of the thermalization of a chaotic quantum system can be defined on the basis of convergence of the projected ensemble to a quantum state design, i.e. a system thermalizes when it becomes indistinguishable, up to the k th moment, from a Haar ensemble of uniformly distributed pure states. Here we consider a random unitary circuit with the brick-wall geometry and analyze its convergence to the Haar ensemble through the frame potential and its mapping to a statistical mechanical problem. This approach allows us to highlight a geometric interpretation of the frame potential based on the existence of a fluctuating membrane, similar to those appearing in the study of entanglement entropies. At large local Hilbert space dimension q , we find that all moments converge simultaneously with a time scaling linearly in the size of region A , a feature previously observed in dual unitary models. However, based on the geometric interpretation, we argue that the scaling at finite q on the basis of rare membrane fluctuations, finding the logarithmic scaling of design times t k = O ( log k ) . Our results are supported with numerical simulations performed at q = 2.
Deep thermalization concerns the emergence of universal pure-state statistics in projected ensembles at late times, yet the mechanism governing the initial growth of randomness remains unclear. Here, we study the short-time dynamics of projected ensembles generated from initially unentangled states, quantifying their randomness using frame potentials. For arbitrary Hamiltonians, provided the initial bath state has full support in the measurement basis, we show that the frame potentials to cubic order in time are determined entirely by the subsystem purity, and hence by the bipartite entanglement generated between the unmeasured subsystem and its complement. The entanglement timescale therefore sets the initial timescale for the growth of local randomness, independently of the bath measurement basis. For unitarily invariant Hamiltonian ensembles, we further relate the projected-ensemble frame potential at order k to the 4k-point spectral form factor, or equivalently to the 2kth frame potential of the global unitary dynamics, establishing a direct connection between local and global randomness. Our results identify entanglement as the mechanism governing the onset of randomness in projected ensembles and clarify how local randomness emerges from global quantum dynamics.
Random many-body states and matrices are both a useful tool to model certain physical systems and an important asset for quantum computation. Realizing 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 and matrices, 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 has been focussed on the case of quantum circuits composed by fully 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. Our results also suggest that the minimally random settings (a) and (b) give an optimal framework to prepare unitary designs, i.e., random unitary matrices.