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Quantum circuits with local unitaries have emerged as a rich playground for the exploration of many-body quantum dynamics of discrete-time systems. While the intrinsic locality makes them particularly suited to run on current quantum processors, the task of verification at non-trivial scales is complicated for non-integrable systems. Here, we study a special class of maximally chaotic circuits known as dual unitary circuits -- exhibiting unitarity in both space and time -- that are known to have exact analytical solutions for certain correlation functions. With advances in noise learning and the implementation of novel error mitigation methods, we show that a superconducting quantum processor with 91 qubits is able to accurately simulate these correlators. We then probe dynamics beyond exact verification, by perturbing the circuits away from the dual unitary point, and compare our results to classical approximations with tensor networks. These results cement error-mitigated digital quantum simulation on pre-fault-tolerant quantum processors as a trustworthy platform for the exploration and discovery of novel emergent quantum many-body phases., Comment: 7 + 23 pages, 3 + 17 figures, 0 + 2 tables

We aim to identify the spatial distribution of vegetation and its growth dynamics with the purpose of obtaining a qualitative assessment of vegetation characteristics tied to its condition, productivity and health, and to land degradation. To do so, we compare a statistical model of vegetation growth and land surface imagery derived vegetation indices. Specifically, we analyze a stochastic cellular automata model and data obtained from satellite images, namely using the Normalized Difference Vegetation Index (NDVI) and the Leaf Area Index (LAI). In the experimental data, we look for areas where vegetation is broken into small patches and qualitatively compare it to the percolating, fragmented, and degraded states that appear in the cellular automata model. We model the periodic effect of seasons, finding numerical evidence of a periodic fragmentation and recovery phenomenology if the model parameters are sufficiently close to the model's percolation transition. We qualitatively recognize these effects in real-world vegetation images and consider them a signal of increased environmental stress and vulnerability. Finally, we show an estimation of the environmental stress in land images by considering both the vegetation density and its clusterization., Comment: 17 pages, 18 figures, the published version

Recent experimental observation of weak ergodicity breaking in Rydberg atom quantum simulators has sparked interest in quantum many-body scars - eigenstates which evade thermalisation at finite energy densities due to novel mechanisms that do not rely on integrability or protection by a global symmetry. A salient feature of some quantum many-body scars is their sub-volume bipartite entanglement entropy. In this work we demonstrate that such exact many-body scars also possess extensive multipartite entanglement structure if they stem from an su(2) spectrum generating algebra. We show this analytically, through scaling of the quantum Fisher information, which is found to be super-extensive for exact scarred eigenstates in contrast to generic thermal states. Furthermore, we numerically study signatures of multipartite entanglement in the PXP model of Rydberg atoms, showing that extensive quantum Fisher information density can be generated dynamically by performing a global quench experiment. Our results identify a rich multipartite correlation structure of scarred states with significant potential as a resource in quantum enhanced metrology., Comment: 6+6 pages, 3+5 figures

Quantum dynamics on quasiperiodic geometries has recently gathered significant attention in ultra-cold atom experiments where non trivial localised phases have been observed. One such quasiperiodic model is the so called Fibonacci model. In this tight-binding model, non-interacting particles are subject to on-site energies generated by a Fibonacci sequence. This is known to induce critical states, with a continuously varying dynamical exponent, leading to anomalous transport. In this work, we investigate whether anomalous diffusion present in the non-interacting system survives in the presence of interactions and establish connections to a possible transition towards a localized phase. We investigate the dynamics of the interacting Fibonacci model by studying real-time spread of density-density correlations at infinite temperature using the dynamical typicality approach. We also corroborate our findings by calculating the participation entropy in configuration space and investigating the expectation value of local observables in the diagonal ensemble., Comment: 12 pages, 10 figures

We numerically study the possibility of many-body localization transition in a disordered quantum dimer model on the honeycomb lattice. By using the peculiar constraints of this model and state-of-the-art exact diagonalization and time evolution methods, we probe both eigenstates and dynamical properties and conclude on the existence of a localization transition, on the available time and length scales (system sizes of up to N=108 sites). We critically discuss these results and their implications., Comment: 26 pages, 16 figures. Revised version

The characterizing feature of a many-body localized phase is the existence of an extensive set of quasi-local conserved quantities with an exponentially localized support. This structure endows the system with the signature logarithmic in time entanglement growth between spatial partitions. This feature differentiates the phase from Anderson localization, in a non-interacting model. Experimentally measuring the entanglement between large partitions of an interacting many-body system requires highly non-local measurements which are currently beyond the reach of experimental technology. In this work we demonstrate that the defining structure of many-body localization can be detected by the dynamics of a simple quantity from quantum information known as the total correlations which is connected to the local entropies. Central to our finding is the necessity to propagate specific initial states, drawn from the Hamiltonian unbiased basis (HUB). The dynamics of the local entropies and total correlations requires only local measurements in space and therefore is potentially experimentally accessible in a range of platforms., Comment: 4 pages, 5 figures, comments are welcome

Investigating many-body localization (MBL) using exact numerical methods is limited by the exponentialgrowth of the Hilbert space. However, localized eigenstates display multifractality and only extend over a vanishing fraction of the Hilbert space. Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to an Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicted, together with a geometrical interpretation of MBL multifractality as a shattering of the Hilbert space., Comment: Special Issue of the Annals of Physics, Localisation 2020, dedicated to Philip W. Anderson

We study the Anderson model on the Bethe lattice by working directly with propagators at real energies $E$. We introduce a novel criterion for the localization-delocalization transition based on the stability of the population of the propagators, and show that it is consistent with the one obtained through the study of the imaginary part of the self-energy. We present an accurate numerical estimate of the transition point, as well as a concise proof of the asymptotic formula for the critical disorder on lattices of large connectivity, as given in [P.W. Anderson 1958]. We discuss how the forward approximation used in analytic treatments of localization problems fits into this scenario and how one can interpolate between it and the correct asymptotic analysis., Comment: Close to published version

We propose a class of mean-field models for the isostatic transition of systems of soft spheres, in which the contact network is modeled as a random graph and each contact is associated to $d$ degrees of freedom. We study such models in the hypostatic, isostatic, and hyperstatic regimes. The density of states is evaluated by both the cavity method and exact diagonalization of the dynamical matrix. We show that the model correctly reproduces the main features of the density of states of real packings and, moreover, it predicts the presence of localized modes near the lower band edge. Finally, the behavior of the density of states $D(\omega)\sim\omega^\alpha$ for $\omega\to 0$ in the hyperstatic regime is studied. We find that the model predicts a nontrivial dependence of $\alpha$ on the details of the coordination distribution., Comment: 15 pages, 14 figures

We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl/. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to $L=26$. This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.

The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics. In analogy to the time average of a classical phase space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper-bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We generalize the total correlations of the diagonal ensemble to more general $\alpha$-Renyi entropies and focus on the the cases $\alpha=1$ and $\alpha=2$ with further numerical extensions in mind. Here we show that the total correlations of the diagonal ensemble is a generic indicator of ergodicity breaking, displaying a sub-extensive behaviour when the system is ergodic. We demonstrate this by investigating its scaling in a range of spin chain models focusing not only on the cases of integrability breaking but also emphasize its role in understanding the transition from an ergodic to a many body localized phase in systems with disorder or quasi-periodicity., Comment: v3: several minor improvements

We study the details of the distribution of the entanglement spectrum (eigenvalues of the reduced density matrix) of a disordered spin chain exhibiting a many-body localization (MBL) transition. In the thermalizing region we identify the evolution under increasing system size of the eigenvalues distribution function, whose thermodynamic limit is close (but possibly different from) the Marchenko-Pastur distribution. From the analysis we extract a correlation length $L_s(h)$ determining the minimum system size to enter the asymptotic region. We find that $L_s(h)$ diverges at the MBL transition. We discuss the nature of the subleading corrections to the entanglement spectrum distribution and to the entanglement entropy., Comment: 15 pages, 6 figures; close to published version

The phenomenon of many-body localization in disordered quantum many-body systems occurs when all transport is suppressed despite the fact that the excitations of the system interact. In this work we report on the numerical simulation of autonomous quantum dynamics for disordered Heisenberg chains when the system is prepared with an initial inhomogeneity in the energy density profile. Using exact diagonalisation and a dynamical code based on Krylov subspaces we are able to simulate dynamics for up to L = 26 spins. We find, surprisingly, the breakdown of energy diffusion even before the many-body localization transition whilst the system is still in the ergodic phase. Moreover, in the ergodic phase we also find a large region in parameter space where the energy dynamics remains diffusive but where spin transport has been previously evidenced to occur only subdiffusively: this is found to be true for initial states composed of infinitely many hydrodynamic modes (square-wave energy profile) or just the single longest mode (sinusoidal profile). This suggestive finding points towards a peculiar ergodic phase where particles are transported slower than energy, reminiscent of the situation in amorphous solids and of the gapped phase of the anisotropic Heisenberg model., Comment: 17 pages, 6 figures, Published version

In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-) or many-body localized phase. This approximation, which consists in summing over the amplitudes of only the shortest paths in the locator expansion, is known to over-estimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is $0.9\%$ off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder $h_c=4.0\pm 0.3$ is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent $\nu=1$. In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium.