All Publications

Here, we focus on the problem of minimizing complex classical cost functions associated with prototypical discrete neural networks, specifically the paradigmatic Hopfield model and binary perceptron. We show that the adiabatic time evolution of QA can be efficiently represented as a suitable Tensor Network. This representation allows for simple classical simulations, well-beyond small sizes amenable to exact diagonalization techniques. We show that the optimized state, expressed as a Matrix Product State (MPS), can be recast into a Quantum Circuit, whose depth scales only linearly with the system size and quadratically with the MPS bond dimension. This may represent a valuable starting point allowing for further circuit optimization on near-term quantum devices.

A large ongoing research effort focuses on variational quantum algorithms (VQAs), representing leading candidates to achieve computational speed-ups on current quantum devices. The scalability of VQAs to a large number of qubits, beyond the simulation capabilities of classical computers, is still debated. Two major hurdles are the proliferation of low-quality variational local minima, and the exponential vanishing of gradients in the cost-function landscape, a phenomenon referred to as barren plateaus. In this work, we show that by employing iterative search schemes, one can effectively prepare the ground state of paradigmatic quantum many-body models, also circumventing the barren plateau phenomenon. This is accomplished by leveraging the transferability to larger system sizes of a class of iterative solutions, displaying an intrinsic smoothness of the variational parameters, a result that does not extend to other solutions found via random-start local optimization. Our scheme could be directly tested on near-term quantum devices, running a refinement optimization in a favorable local landscape with nonvanishing gradients.

We consider the two-dimensional quantum Ising model, in absence of disorder, subject to periodic imperfect global spin flips. We show by a combination of exact diagonalization and tensor-network methods that the system can sustain a spontaneously broken discrete time-translation symmetry. Employing careful scaling analysis, we show the feasibility of a two-dimensional discrete time-crystal (DTC) prethermal phase. Despite an unbounded energy pumped into the system, in the high-frequency limit, a well-defined effective Hamiltonian controls a finite-temperature intermediate regime, wherein local time averages are described by thermal averages. As a consequence, the long-lived stability of the DTC relies on the existence of a long-range ordered phase at finite temperature. Interestingly, even for large deviations from the perfect spin flip, we observe a nonperturbative change in the decay rate of the order parameter, which is related to the long-lived stability of the magnetic domains in 2D.

We study the dynamics of the quantum Dyson hierarchical model in its paramagnetic phase. An initial state made by a local excitation of the paramagnetic ground state is considered. We provide analytical predictions for its time evolution, solving the single-particle dynamics on a hierarchical network. A localization mechanism is found, and the excitation remains close to its initial position at arbitrary times. Furthermore, a universal scaling among space and time is found that is related to the algebraic decay of the interactions as r-1-σ. We compare our predictions to numerics, employing tensor network techniques, for large magnetic fields, discussing the robustness of the mechanism in the full many-body dynamics.

By taking inspiration from the backflow transformation for correlated systems, we introduce a tensor network Ansatz which extends the well-established matrix product state representation of a quantum many-body wave function. This structure provides enough resources to ensure that states in dimensions larger than or equal to one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte Carlo algorithms. We benchmark the Ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging S=1/2 two-dimensional (2D) J1-J2 model, demonstrating that it is competitive with the state-of-the-art methods in 2D.

Discrete time crystals represent a paradigmatic nonequilibrium phase of periodically driven matter. Protecting its emergent spatiotemporal order necessitates a mechanism that hinders the spreading of defects, such as localization of domain walls in disordered quantum spin chains. In this work, we establish the effectiveness of a different mechanism arising in clean spin chains: the confinement of domain walls into "mesonic"bound states. We consider translationally invariant quantum Ising chains periodically kicked at arbitrary frequency, and we discuss two possible routes to domain-wall confinement: longitudinal fields and interactions beyond nearest neighbors. We study the impact of confinement on the order-parameter evolution by constructing domain-wall-conserving effective Hamiltonians and analyzing the resulting dynamics of domain walls. On the one hand, we show that for arbitrary driving frequency, the symmetry-breaking-induced confining potential gets effectively averaged out by the drive, leading to deconfined dynamics. On the other hand, we rigorously prove that increasing the range R of spin-spin interactions Ji,j beyond nearest neighbors enhances the order-parameter lifetime exponentially in R. Our theory predictions are corroborated by a combination of exact and matrix-product-state simulations for finite and infinite chains, respectively. The long-lived stability of spatiotemporal order identified in this work does not rely on Floquet prethermalization nor on eigenstate order, but rather on the nonperturbative origin of vacuum-decay processes. We point out the experimental relevance of this new mechanism for stabilizing a long-lived time-crystalline response in Rydberg-dressed spin chains.

We develop a general approach to compute the symmetry-resolved Rényi and von Neumann entanglement entropies (SREE) of thermodynamic macrostates in interacting integrable systems. Our method is based on a combination of the thermodynamic Bethe ansatz and the Gärtner-Ellis theorem from large deviation theory. We derive an explicit simple formula for the von Neumann SREE, which we show to coincide with the thermodynamic Yang-Yang entropy of an effective macrostate determined by the charge sector. Focusing on the XXZ Heisenberg spin chain, we test our result against iTEBD calculations for thermal states, finding good agreement. As an application, we provide analytic predictions for the asymptotic value of the SREE following a quantum quench.

We study the relaxation of the local ferromagnetic order in the transverse field quantum Ising chain with power-law decaying interactions ∼ 1/r^α. We prepare the system in the GHZ state and study the time evolution of the probability distribution function (PDF) of the order parameter within a block of when quenching the transverse field. The model is known to support long range order at finite temperature for α ≤ 2.0. In this regime, quasi-localized topological magnetic defects are expected to strongly affect the equilibration of the full probability distribution. We highlight different dynamical regimes where gaussification mechanism may be slowed down by confinement and eventually breaks. We further study the PDF dynamics induced by changing the effective dimensionality of the system; we mimic this by quenching the range of the interactions. As a matter of fact, the behavior of the system crucially depends on the value of α governing the unitary evolution.

Nonequilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce an abrupt change in the scaling law of the bipartite entanglement entropy, thus suggesting the existence of different nonequilibrium regimes. However, our understanding of how these regimes appear and whether they survive in the thermodynamic limit is much less established. Here we investigate these questions on a one-dimensional quadratic fermionic model: this allows us to reach system sizes relevant in the thermodynamic sense. We show that local projective measurements induce a qualitative modification of the time growth of the entanglement entropy which changes from linear to logarithmic. However, in the stationary regime, the logarithmic behavior of the entanglement entropy does not survive in the thermodynamic limit and, for any finite value of the measurement rate, we numerically show the existence of a single area-law phase for the entanglement entropy. Finally, exploiting the quasiparticle picture, we further support our results by analyzing the fluctuations of the stationary entanglement entropy and its scaling behavior.

In many cases, neural networks can be mapped into tensor networks with an exponentially large bond dimension. Here, we compare different sub-classes of neural network states, with their mapped tensor network counterpart for studying the ground state of short-range Hamiltonians. We show that when mapping a neural network, the resulting tensor network is highly constrained and thus the neural network states do in general not deliver the naive expected drastic improvement against the state-of-the-art tensor network methods. We explicitly show this result in two paradigmatic examples, the 1D ferromagnetic Ising model and the 2D antiferromagnetic Heisenberg model, addressing the lack of a detailed comparison of the expressiveness of these increasingly popular, variational ansätze.

We present an unconstrained tree-tensor-network approach to the study of lattice gauge theories in two spatial dimensions, showing how to perform numerical simulations of theories in the presence of fermionic matter and four-body magnetic terms, at zero and finite density, with periodic and open boundary conditions. We exploit the quantum-link representation of the gauge fields and demonstrate that a fermionic rishon representation of the quantum links allows us to efficiently handle the fermionic matter while finite densities are naturally enclosed in the tensor network description. We explicitly perform calculations for quantum electrodynamics in the spin-one quantum-link representation on lattice sizes of up to 16×16 sites, detecting and characterizing different quantum regimes. In particular, at finite density, we detect signatures of a phase separation as a function of the bare mass values at different filling densities. The presented approach can be extended straightforwardly to three spatial dimensions.

In spin chains with local unitary evolution preserving the magnetization Sz, the domain-wall state typically "melts."At large times, a nontrivial magnetization profile develops in an expanding region around the initial position of the domain wall. For nonintegrable dynamics, the melting is diffusive, with entropy production within a melted region of size t. In contrast, when the evolution is integrable, ballistic transport dominates and results in a melted region growing linearly in time, with no extensive entropy production: The spin chain remains locally in states of zero entropy at any time. Here we show that, for the integrable spin-1/2 XXZ chain, low-energy quantum fluctuations in the melted region give rise to an emergent Luttinger liquid which, remarkably, differs from the equilibrium one. The striking feature of this emergent Luttinger liquid is its quasiparticle charge (or Luttinger parameter K), which acquires a fractal dependence on the XXZ chain anisotropy parameter Δ.

We study the relaxation of the local ferromagnetic order in the quantum Ising chain in a slant field with both longitudinal and transverse components. After preparing the system in a fully polarised state, we analyse the time evolution of the entire probability distribution function (PDF) of the magnetisation within a block of l spins. We first analyse the effect of confinement on the Gaussification of the PDF for large l, showing that the melting of initial order is suppressed when the longitudinal field is aligned to initial magnetisation while it is speed up when it is in the opposite direction. Then we study the thermalisation dynamics. In the paramagnetic region, the PDF quickly shows thermal features. Conversely, in the ferromagnetic phase, when confinement takes place, the relaxation suffers a typical slowing down which depends on the interplay between the strength of the longitudinal field, the density of excitations, and the direction of the initial polarisation. Even when the initial magnetisation is aligned oppositely to the longitudinal field, confinement prevents thermalisation in the accessible timescale, as it is neatly bared by the PDF.

Confinement of excitations induces quasilocalized dynamics in disorder-free isolated quantum many-body systems in one spatial dimension. This occurrence is signaled by severe suppression of quantum correlation spreading and of entanglement growth, long-time persistence of spatial inhomogeneities, and long-lived coherent oscillations of local observables. In this work, we present a unified understanding of these dramatic effects. The slow dynamical behavior is shown to be related to the Schwinger effect in quantum electrodynamics. We demonstrate that it is quantitatively captured for long-time scales by effective Hamiltonians exhibiting Stark localization of excitations and weak growth of the entanglement entropy for arbitrary coupling strength. This analysis explains the phenomenology of real-time string dynamics investigated in a number of lattice gauge theories, as well as the anomalous dynamics observed in quantum Ising chains after quenches. Our findings establish confinement as a robust mechanism for hindering the approach to equilibrium in translationally invariant quantum statistical systems with local interactions.

We show how to implement a Rydberg-atom quantum simulator to study the nonequilibrium dynamics of an Abelian (1+1)-dimensional lattice gauge theory. The implementation locally codifies the degrees of freedom of a Z3 gauge field, once the matter field is integrated out by means of the Gauss local symmetries. The quantum simulator scheme is based on currently available technology and thus is scalable to considerable lattice sizes. It allows, within experimentally reachable regimes, us to explore different string dynamics and to infer information about the Schwinger U(1) model.

The quasi-particle picture is a powerful tool to understand the entanglement spreading in many-body quantum systems after a quench. As an input, the structure of the excitations’ pattern of the initial state must be provided, the common choice being pairwise-created excitations. However, several cases exile this simple assumption. In this work we investigate weakly-interacting to free quenches in one dimension. This results in a far richer excitations’ pattern where multiplets with a larger number of particles are excited. We generalize the quasi-particle ansatz to such a wide class of initial states, providing a small-coupling expansion of the Rényi entropies. Our results are in perfect agreement with iTEBD numerical simulations.

We exploit the knowledge of the entanglement spectrum in the ground state of the gapped XXZ spin chain to derive asymptotic exact results for the full counting statistics of the transverse magnetisation in a large spin block of length ℓ. We found that for a subsystem of even length the full counting statistics is Gaussian, while for odd subsystems it is the sum of two Gaussian distributions. We test our analytic predictions with accurate tensor networks simulations. As a byproduct, we also obtain the symmetry (magnetisation) resolved entanglement entropies.

Injecting a sufficiently large energy density into an isolated many-particle system prepared in a state with long-range order will lead to the melting of the order over time. Detailed information about this process can be derived from the quantum mechanical probability distribution of the order parameter. We study this process for the paradigmatic case of the spin-1/2 Heisenberg XXZ chain. We determine the full quantum mechanical distribution function of the staggered subsystem magnetization as a function of time after a quantum quench from the classical Néel state. We establish the existence of an interesting regime at intermediate times that is characterized by a very broad probability distribution. Based on our findings we propose a simple general physical picture of how long-range order melts.

We study the out-of-equilibrium probability distribution function of the local order parameter in the transverse field Ising quantum chain. Starting from a fully polarised state, the relaxation of the ferromagnetic order is analysed: we obtain a full analytical description of the late-time stationary distribution by means of a remarkable relation to the partition function of a 3-states classical model. Accordingly, depending on the phase whereto the post-quench Hamiltonian belongs, the probability distribution may locally retain memories of the initial long-range order. When quenching deep in the broken-symmetry phase, we show that the stationary order-parameter statistics is indeed related to that of the ground state. We highlight this connection by inspecting the ground-state equilibrium properties, where we propose an effective description based on the block-diagonal approximation of the n-point spin correlation functions.

Transport phenomena are central to physics, and transport in the many-body and fully-quantum regime is attracting an increasing amount of attention. It has been recently revealed that some quantum spin chains support ballistic transport of excitations at all energies. However, when joining two semi-infinite ballistic parts, such as the XX and XXZ spin-1/2 models, our understanding suddenly becomes less established. Employing a matrix-product-state ansatz of the wavefunction, we study the relaxation dynamics in this latter case. Here we show that it takes place inside a light cone, within which two qualitatively different regions coexist: an inner one with a strong tendency towards thermalization, and an outer one supporting ballistic transport. We comment on the possibility that even at infinite time the system supports stationary currents and displays a non-zero Kapitza boundary resistance. Our study paves the way to the analysis of the interplay between transport, integrability, and local defects.