##### Density of states of the lattice Schwinger model

I. Papaefstathiou, D. Robaina, J.I. Cirac, M.C. Bañuls

Physical Review D 104, 014514 (2021).

Using a recently introduced tensor network method, we study the density of states of the lattice Schwinger model, a standard testbench for lattice gauge theory numerical techniques, but also the object of recent experimental quantum simulations. We identify regimes of parameters where the spectrum appears to be symmetric and displays the expected continuum properties even for finite lattice spacing and number of sites. However, we find that for moderate system sizes and lattice spacing of ga similar to 0(1), the spectral density can exhibit very different properties with a highly asymmetric form. We also explore how the method can be exploited to extract thermodynamic quantities.

##### Optimal sampling of dynamical large deviations via matrix product states

L. Causer, M.C. Banuls, J. P. Garrahan

Physical Review E 103, 62144 (2021).

The large deviation (LD) statistics of dynamical observables is encoded in the spectral properties of deformed Markov generators. Recent works have shown that tensor network methods are well suited to compute the relevant leading eigenvalues and eigenvectors accurately. However, the efficient generation of the corresponding rare trajectories is a harder task. Here we show how to exploit the MPS approximation of the dominant eigenvector to implement an efficient sampling scheme which closely resembles the optimal (so-called "Doob") dynamics that realises the rare events. We demonstrate our approach on three well-studied lattice models, the Fredrickson-Andersen and East kinetically constrained models (KCMs), and the symmetric simple exclusion process (SSEP). We discuss how to generalise our approach to higher dimensions.

##### Tensors cast their nets for quarks

M.C. Bañuls, K. Cichy

nature physics, News & views 17, 762–763 (2021).

Many aspects of gauge theories — such as the one underlying quantum chromodynamics, which describes quark physics — evade common numerical methods. Tensor networks are getting closer to a solution, having successfully tackled the related problem of a three-dimensional quantum link model.

##### Rényi free energy and variational approximations to thermal states

G. Giudice, A. Cakan, J.I. Cirac, M.C. Banuls

Physical Review B 103, 205128 (2021).

We propose the construction of thermodynamic ensembles that minimize the Rényi free energy, as an alternative to Gibbs states. For large systems, the local properties of these Rényi ensembles coincide with those of thermal equilibrium, and they can be used as approximations to thermal states. We provide algorithms to find tensor network approximations to the 2-Rényi ensemble. In particular, a matrix-product-state representation can be found by using gradient-based optimization on Riemannian manifolds, or via a non-linear evolution which yields the desired state as a fixed point. We analyze the performance of the algorithms and the properties of the ensembles on one-dimensional spin chains.

##### Algorithms for quantum simulation at finite energies

S. Lu, M.C. Banuls, J.I. Cirac

Physical Review X Quantum 2, 20321 (2021).

We introduce two kinds of quantum algorithms to explore microcanonical and canonical properties of many-body systems. The first one is a hybrid quantum algorithm that, given an efficiently preparable state, computes expectation values in a finite energy interval around its mean energy. This algorithm is based on a filtering operator, similar to quantum phase estimation, which projects out energies outside the desired energy interval. However, instead of performing this operation on a physical state, it recovers the physical values by performing interferometric measurements without the need to prepare the filtered state. We show that the computational time scales polynomially with the number of qubits, the inverse of the prescribed variance, and the inverse error. In practice, the algorithm does not require the evolution for long times, but instead a significant number of measurements in order to obtain sensible results. Our second algorithm is a quantum-assisted Monte Carlo sampling method to compute other quantities which approach the expectation values for the microcanonical and canonical ensembles. Using classical Monte Carlo techniques and the quantum computer as a resource, this method circumvents the sign problem that is plaguing classical Quantum Monte Carlo simulations, as long as one can prepare states with suitable energies. All algorithms can be used with small quantum computers and analog quantum simulators, as long as they can perform the interferometric measurements. We also show that this last task can be greatly simplified at the expense of performing more measurements.

##### Approximating the long time average of the density operator: Diagonal ensemble

A. Cakan, J.I. Cirac, M.C. Banuls

Physical Review B 103, 115113 (2021).

For an isolated generic quantum system out of equilibrium, the long time average of observables is given by the diagonal ensemble, i.e. the mixed state with the same probability for energy eigenstates as the initial state but without coherences between different energies. In this work we present a method to approximate the diagonal ensemble using tensor networks. Instead of simulating the real time evolution, we adapt a filtering scheme introduced earlier in [Phys. Rev. B 101, 144305 (2020)] to this problem. We analyze the performance of the method on a non-integrable spin chain, for which we observe that local observables converge towards thermal values polynomially with the inverse width of the filter.

##### Simulating 2+1D Z(3) Lattice Gauge Theory with an Infinite Projected Entangled-Pair State

D. Robaina, M.C. Banuls, J.I. Cirac

Physical Review Letters 126 (5), 050401 (2021).

We simulate a zero-temperature pure Z(3) lattice gauge theory in 2 + 1 dimensions by using an iPEPS (infmite projected entangled-pair state) Ansatz for the ground state. Our results are therefore directly valid in the thermodynamic limit. They clearly show two distinct phases separated by a phase transition. We introduce an update strategy that enables plaquette terms and Gauss-law constraints to be applied as sequences of two-body operators. This allows the use of the most up-to-date iPEPS algorithms. From the calculation of spatial Wilson loops we are able to prove the existence of a confined phase. We show that with relatively low computational cost it is possible to reproduce crucial features of gauge theories. We expect that the strategy allows the extension of iPEPS studies to more general LGTs.

##### Dynamics and large deviation transitions of the XOR-Fredrickson-Andersen kinetically constrained model

L. Causer, I. Lesanovsky, M.C. Banuls, J.P. Garrahan

Physical Review E 102 (5), 052132 (2020).

We study a one-dimensional classical stochastic kinetically constrained model (KCM) inspired by Rydberg atoms in their "facilitated" regime, where sites can flip only if a single of their nearest neighbors is excited. We call this model "XOR-FA" to distinguish it from the standard Fredrickson-Andersen (FA) model. We describe the dynamics of the XOR-FA model, including its relation to simple exclusion processes in its domain wall representation. The interesting relaxation dynamics of the XOR-FA is related to the prominence of large dynamical fluctuations that lead to phase transitions between active and inactive dynamical phases as in other KCMs. By means of numerical tensor network methods we study in detail such transitions in the dynamical large deviation regime.

##### Variational Monte Carlo simulation with tensor networks of a pure Z(3) gauge theory in (2+1)D

P. Emonts, M.C. Banuls, I. Cirac, E. Zohar

Physical Review D 102 (7), 074501 (2020).

Variational minimization of tensor network states enables the exploration of low energy states of lattice gauge theories. However, the exact numerical evaluation of high-dimensional tensor network states remains challenging in general. In [E. Zohar and J. I. Cirac, Phys. Rev. D 97, 034510 (2018)] it was shown how, by combining gauged Gaussian projected entangled pair states with a variational Monte Carlo procedure, it is possible to efficiently compute physical observables. In this paper we demonstrate how this approach can be used to investigate numerically the ground state of a lattice gauge theory. More concretely, we explicitly carry out the variational Monte Carlo procedure based on such contraction methods for a pure gauge KogutSusskind Hamiltonian with a Z(3) gauge field in two spatial dimensions. This is a first proof of principle to the method, which provides an inherent way to increase the number of variational parameters and can be readily extended to systems with physical fermions.

##### Phase structure and real-time dynamics of the massive Thirring model in 1+1 dimensions using the tensor-network method

M.C. Banuls, K. Cichy, H.T. Hung, Y.J. Kao, D. Lin, Y.P. Lin, D.T.L. Tan

Proceedings of Science LATTICE2019, 22 (2020).

We present concluding results from our study for zero-temperature phase structure of the massive Thirring model in 1+1 dimensions with staggered regularisation. Employing the method of matrix product states, several quantities, including two types of correlators, are investigated, leading to numerical evidence of a Berezinskii-Kosterlitz-Thouless phase transition. Exploratory results for real-time dynamics pertaining to this transition, obtained using the approaches of variational uniform matrix product state and time-dependent variational principle, are also discussed.

##### From spin chains to real-time thermal field theory using tensor networks

M.C. Bañuls, M. P. Heller, K. Jansen, J. Knaute, and V. Svensson

Physical Review Research 2, 33301 (2020).

One of the most interesting directions in theoretical high-energy and condensed-matter physics is understanding dynamical properties of collective states of quantum field theories. The most elementary tool in this quest is retarded equilibrium correlators governing the linear response theory. In this article we examine tensor networks as a way of determining them in a fully ab initio way in a class of (1+1)-dimensional quantum field theories arising as infrared descriptions of quantum Ising chains.We show that, complemented with signal analysis using the Prony method, tensor network calculations for intermediate times provide a powerful way to explore the structure of singularities of the correlator in the complex frequency plane and to make predictions about the thermal response to perturbations in a class of nonintegrable interacting quantum field theories.

##### Simulating lattice gauge theories within quantum technologies

M.C. Banuls, R. Blatt, J. Catani, A. Celi, J.I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S: Montangero, C.A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U.J. Wiese, M. Wingate, K. Zakrzewski, P. Zoller

European Physical Journal D 47 (8), 165 (2020).

Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented - a classical simulation approach - applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed.

##### Prethermalization of quantum systems interacting with non-equilibrium environments

A. Angles-Castillo, M.C. Banuls, A. Perez, I. De Vega

New Journal of Physics 22 (8), 083067 (2020).

The usual paradigm of open quantum systems falls short when the environment is actually coupled to additional fields or components that drive it out of equilibrium. Here we explore the simplest such scenario, by considering a two level system coupled to a first thermal reservoir that in turn couples to a second thermal bath at a different temperature. We derive a master equation description for the system and show that, in this situation, the dynamics can be especially rich. In particular, we observe prethermalization, a transitory phenomenon in which the system initially approaches thermal equilibrium with respect to the first reservoir, but after a longer time converges to the thermal state dictated by the temperature of the second environment. Using analytical arguments and numerical simulations, we analyze the occurrence of this phenomenon, and how it depends on temperatures and coupling strengths. The phenomenology gets even richer if the system is placed between two such non-equilibrium environments. In this case, the energy current through the system may exhibit transient features and even switch direction, before the system eventually reaches a non-equilibrium steady state.

##### Quantum East Model: Localization, Nonthermal Eigenstates, and Slow Dynamics

N. Pancotti, G. Giudice, J.I. Cirac, J.P. Garrahan, M.C. Banuls

Physical Review X 10 (2), 021051 (2020).

We study in detail the properties of the quantum East model, an interacting quantum spin chain inspired by simple kinetically constrained models of classical glasses. Through a combination of analytics, exact diagonalization, and tensor-network methods, we show the existence of a transition, from a fast to a slow thermalization regime, which manifests itself throughout the spectrum. On the slow side, by exploiting the localization of the ground state and the form of the Hamiltonian, we explicitly construct a large (exponential in size) number of nonthennal states that become exact finite-energy-density eigenstates in the large size limit, as expected for a true phase transition. A "superspin" generalization allows us to fmd a further large class of area-law states proved to display very slow relaxation. These states retain memory of their initial conditions for extremely long times. Our numerical analysis reveals that the localization properties are not limited to the ground state and that many eigenstates have large overlap with product states and can be approximated well by matrix product states at arbitrary energy densities. The mechanism that induces localization to the ground state, and hence the nonthermal behavior of the system, can be extended to a wide range of models including a number of simple spin chains. We discuss implications of our results for slow thermalization and nonergodicity more generally in disorder-free systems with constraints, and we give numerical evidence that these results may be extended to two-dimensional systems.

##### Entanglement and its relation to energy variance for local one-dimensional Hamiltonians

M.C. Banuls, D.A. Huse, J.I. Cirac

Physical Review B 101 (14), 144305 (2020).

We explore the relation between the entanglement of a pure state and its energy variance for a local one-dimensional Hamiltonian, as the system size increases. In particular, we introduce a construction which creates a matrix product state of arbitrarily small energy variance delta(2) for N spins, with bond dimension scaling as root ND01/delta, where D-0 > 1 is a constant. This implies that a polynomially increasing bond dimension is enough to construct states with energy variance that vanishes with the inverse of the logarithm of the system size. We run numerical simulations to probe the construction on two different models and compare the local reduced density matrices of the resulting states to the corresponding thermal equilibrium. Our results suggest that the spatially homogeneous states with logarithmically decreasing variance, which can be constructed efficiently, do converge to the thermal equilibrium in the thermodynamic limit, while the same is not true if the variance remains constant.

##### Markovianity of an emitter coupled to a structured spin-chain bath

J. Roos, J.I. Cirac, M.C. Banuls

Physical Review A 101 (4), 042114 (2020).

We analyze the dynamics of a spin-1/2 subsystem coupled to a spin chain. We simulate numerically the full quantum many-body system for various sets of parameters and initial states of the chain, and characterize the divisibility of the subsystem dynamics, i.e., whether it is Markovian and can be described by a (time-dependent) master equation. We identify regimes in which the subsystem admits such Markovian description, despite the many-body setting, and provide insight about why the same is not possible in other regimes. Interestingly, coupling the subsystem at the edge, instead of the center, of the chain gives rise to qualitatively distinct behavior.

##### Probing Thermalization through Spectral Analysis with Matrix Product Operators

Y.L. Yang, S. Iblisdir, J.I. Cirac, M.C. Banuls

Physical Review Letters 124 (10), 100602 (2020).

We combine matrix product operator techniques with Chebyshev polynomial expansions and present a method that is able to explore spectral properties of quantum many-body Hamiltonians. In particular, we show how this method can be used to probe thermalization of large spin chains without explicitly simulating their time evolution, as well as to compute full and local densities of states. The performance is illustrated with the examples of the Ising and PXP spin chains. For the nonintegrable Ising chain, our findings corroborate the presence of thermalization for several initial states, well beyond what direct time-dependent simulations have been able to achieve so far.

##### Review on novel methods for lattice gauge theories

M.C. Bañuls, K. Cichy

Reports on Progress in Physics 83 (2), 024401 (2020).

Formulating gauge theories on a lattice offers a genuinely non-perturbative way of studying quantum field theories, and has led to impressive achievements. In particular, it significantly deepened our understanding of quantum chromodynamics. Yet, some very relevant problems remain inherently challenging, such as real time evolution, or the presence of a chemical potential, cases in which Monte Carlo simulations are hindered by a sign problem.

In the last few years, a number of possible alternatives have been put forward, based on quantum information ideas, which could potentially open the access to areas of research that have so far eluded more standard methods. They include tensor network calculations, quantum simulations with different physical platforms and quantum computations, and constitute nowadays a vibrant research area. Experts from different fields, including experimental and theoretical high energy physics, condensed matter, and quantum information, are turning their attention to these interdisciplinary possibilities, and driving the progress of the field. The aim of this article is to review the status and perspectives of these new avenues for the exploration of lattice gauge theories.

##### Phase structure of the (1+1)-dimensional massive Thirring model from matrix product states

M.C. Bañuls, K. Cichy, Y. Kao, D.Lin, Y. Lin, D. Tan

Physical Review D Physical Review D, 94504 (2019).

Employing matrix product states as an ansatz, we study the nonthermal phase structure of the (1 + 1)-dimensional massive Thirring model in the sector of a vanishing total fermion number with staggered regularization. In this paper, details of the implementation for this project are described. To depict the phase diagram of the model, we examine the entanglement entropy, the fermion bilinear condensate, and two types of correlation functions. Our investigation shows the existence of two phases, with one of them being critical and the other gapped. An interesting feature of the phase structure is that the theory with the nonzero fermion mass can be conformal. We also find clear numerical evidence that these phases are separated by a transition of the Berezinskii-Kosterlitz-Thouless type. Results presented in this paper establish the possibility of using the matrix product states for probing this type of phase transition in quantum field theories. They can provide information for further exploration of scaling behavior, and they serve as an important ingredient for controlling the continuum extrapolation of the model.

##### Using Matrix Product States to Study the Dynamical Large Deviations of Kinetically Constrained Models

M.C. Banuls, J.P. Garrahan

Physical Review Letters 123 (20), 200601 (2019).

Here we demonstrate that tensor network techniques-originally devised for the analysis of quantum many-body problems-are well suited for the detailed study of rare event statistics in kinetically constrained models (KCMs). As concrete examples, we consider the Fredrickson-Andersen and East models, two paradigmatic KCMs relevant to the modeling of glasses. We show how variational matrix product states allow us to numerically approximate-systematically and with high accuracy-the leading eigenstates of the tilted dynamical generators, which encode the large deviation statistics of the dynamics. Via this approach, we can study system sizes beyond what is possible with other methods, allowing us to characterize in detail the finite size scaling of the trajectory-space phase transition of these models, the behavior of spectral gaps, and the spatial structure and "entanglement" properties of dynamical phases. We discuss the broader implications of our results.

##### Tensor Networks and their use for Lattice Gauge Theories

M.C. Banuls, K. Cichy, J.I. Cirac, K. Jansen, S. Kühn

Proceedings of Science LATTICE2018, 22 (2019).

Tensor Network States are ansaetze for the efficient description of quantum many-body systems. Their success for one dimensional problems, together with the fact that they do not suffer from the sign problem and can address the simulation of real time evolution, have turned them into one of the most promising techniques to study strongly correlated systems.In the realm of Lattice Gauge Theories they can offer an alternative to standard lattice Monte Carlo calculations, which are suited for static properties and regimes where no sign problem appears. The application of Tensor Networks to this kind of problems is a young but rapidly evolving research field. This paper reviews some of the recent progress in this area, and how, using one dimensional models as testbench, some fundamental milestones have been reached that may pave the way to more ambitious goals.

##### Investigation of the 1+1 dimensional Thirring model using the method of matrix product states

M.C. Banuls, K. Cichy, Y.J. Kao, C.J.D. Lin, Y.P. Lin, T.L. Tan

Proceedings of Science LATTICE2018, 229 (2019).

We present preliminary results of a study on the non-thermal phase structure of the (1+1) dimensional massive Thirring model, employing the method of matrix product states. Through investigating the entanglement entropy, the fermion correlators and the chiral condensate, it is found that this approach enables us to observe numerical evidence of a Kosterlitz-Thouless phase transition in the model.

##### Gaussian states for the variational study of (1+1)-dimensional lattice gauge models

P. Sala, T. Shi, S. Kuehn, M.C. Banuls, E. Demler, J.I. Cirac

Proceedings of Science LATTICE2018, 230 (2019).

We introduce a variational ansatz based on Gaussian states for (1+1)-dimensional lattice gauge models. To this end we identify a set of unitary transformations which decouple the gauge degrees of freedom from the matter fields. Using our ansatz, we study static aspects as well as real-time dynamics of string breaking in two (1+1)-dimensional theories, namely QED and two-color QCD. We show that our ansatz captures the relevant features and is in excellent agreement with data from numerical calculations with tensor networks.