##### 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.

##### Quantum algorithms for powering stable Hermitian matrices

G. González, R. Trivedi, J.I. Cirac

Physical Review A 103, 062420 (2021).

Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering and underlies the solution of time-homogeneous linear ordinary differential equations, simulation of discrete-time Markov chains, or discovering the spectral properties of matrices with iterative methods. In this paper, we investigate the possibility of speeding up matrix powering of sparse stable Hermitian matrices on a quantum computer. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms: (i) a fast-forwarding algorithm that builds on construction of Apers and Sarlette [Quantum Inf. Comput. 19, 181 (2019)] and (ii) an algorithm based on Hamiltonian simulation. Furthermore, by mapping the N-bit parity determination problem to a matrix powering problem, we provide no-go theorems that limit the quantum speedups achievable in powering non-Hermitian matrices.

##### 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.

##### Quantum coherence as a signature of chaos

N. Anand, G. Styliaris, M. Kumari, P. Zanardi

Physical Review Research 3, 023214 (2021).

We establish a rigorous connection between quantum coherence and quantum chaos by employing coherence measures originating from the resource theory framework as a diagnostic tool for quantum chaos. We quantify this connection at two different levels: quantum states and quantum channels. At the level of states, we show how several well-studied quantifiers of chaos are, in fact, quantum coherence measures in disguise (or closely related to them). We further this connection for all quantum coherence measures by using tools from majorization theory. Then we numerically study the coherence of chaotic-versus-integrable eigenstates and find excellent agreement with random matrix theory in the bulk of the spectrum. At the level of channels, we show that the coherence-generating power (CGP)—a measure of how much coherence a dynamical process generates on average—emerges as a subpart of the out-of-time-ordered correlator (OTOC), a measure of information scrambling in many-body systems. Via numerical simulations of the (nonintegrable) transverse-field Ising model, we show that the OTOC and CGP capture quantum recurrences in quantitatively the same way. Moreover, using random matrix theory, we analytically characterize the OTOC-CGP connection for the Haar and Gaussian ensembles. In closing, we remark on how our coherence-based signatures of chaos relate to other diagnostics, namely, the Loschmidt echo, OTOC, and the Spectral Form Factor.

##### Universal signatures of Dirac fermions in entanglement and charge fluctuations

V. Crépel, A. Hackenbroich, N. Regnault, B. Estienne

Physical Review B 103, 235108 (2021).

We investigate the entanglement entropy (EE) and charge fluctuations in models where the low-energy physics is governed by massless Dirac fermions. We focus on the response to flux insertion which, for the EE, is widely assumed to be universal, i.e., independent of the microscopic details. We provide an analytical derivation of the EE and charge fluctuations for the seminal example of graphene, using the dimensional reduction of its tight-binding model to the one-dimensional Su-Schrieffer-Heeger model. Our asymptotic expression for the EE matches the conformal field theory prediction. We show that the charge variance has the same asymptotic behavior, up to a constant prefactor. To check the validity of universality arguments, we numerically consider several models, with different geometries and numbers of Dirac cones, and either for strictly two-dimensional models or for a gapless surface mode of three-dimensional topological insulators. We also show that the flux response does not depend on the entangling surface geometry as long as it encloses the flux. Finally we consider the universal corner contributions to the EE. We show that in the presence of corners, the Kitaev-Preskill subtraction scheme provides nonuniversal, geometry-dependent results.

##### Generating function for tensor network diagrammatic summation

W.L. Tu, H.K. Wu, N. Schuch, N. Kawashima, J.Y. Chen

Physical Review B 103, 205155 (2021).

The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among others, excitation spectrum and long-range interacting systems can be studied using TNs, where one however confronts the intricate summation over an extensive number of tensor diagrams. Here, we introduce a set of generating functions, which encode the diagrammatic summations as leading-order series expansion coefficients. Combined with automatic differentiation, the generating function allows us to solve the problem of TN diagrammatic summation. We illustrate this scheme by computing variational excited states and the dynamical structure factor of a quantum spin chain, and further investigating entanglement properties of excited states. Extensions to infinite-size systems and higher dimension are outlined.

##### 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.

##### Preparation and verification of tensor network states

E. Cruz, F. Baccari, J. Tura, N. Schuch, J.I. Cirac

We consider a family of tensor network states defined on regular lattices that can be efficiently prepared with a quantum computer by adiabatic evolution. This family comprises relevant classes of states, such as injective Matrix Product and Projected Entangled-Pair States, and some corresponding to classical spin models. We show how uniform lower bounds to the gap of the parent Hamiltonian along the adiabatic trajectory can be efficiently computed using semi-definite programming. We also derive a set of observables whose expectation values can be easily determined and that form a complete set, in the sense that they uniquely characterize the state. We identify a subset of those observables which can be efficiently computed if one has access to the quantum state and local measurements, and analyze how they can be used in verification procedures.

##### Localizable quantum coherence

A. Hamma, G. Styliaris, P. Zanardi

Physics Letters A 397, 127264 (2021).

Coherence is a fundamental notion in quantum mechanics, defined relative to a reference basis. As such, it does not necessarily reveal the locality of interactions nor takes into account the accessible operations in a composite quantum system. In this paper, we put forward a notion of localizable coherence as the coherence that can be stored in a particular subsystem, either by measuring or just by disregarding the rest. We examine its spreading, its average properties in the Hilbert space and show that it can be applied to reveal the real-space structure of states of interest in quantum many-body theory, for example, localized or topological states.

##### Generalization of group-theoretic coherent states for variational calculations

T. Guaita, L. Hackl, T. Shi, E. Demler, J.I. Cirac

Physical Review Research 3, 023090 (2021).

We introduce families of pure quantum states that are constructed on top of the well-known Gilmore-Perelomov group-theoretic coherent states. We do this by constructing unitaries as the exponential of operators quadratic in Cartan subalgebra elements and by applying these unitaries to regular group-theoretic coherent states. This enables us to generate entanglement not found in the coherent states themselves, while retaining many of their desirable properties. Most importantly, we explain how the expectation values of physical observables can be evaluated efficiently. Examples include generalized spin-coherent states and generalized Gaussian states, but our construction can be applied to any Lie group represented on the Hilbert space of a quantum system. We comment on their applicability as variational families in condensed matter physics and quantum information.

##### Weakly invasive metrology: quantum advantage and physical implementations

M. Perarnau-Llobet, D. Malz, J.I. Cirac

Quantum 5, 446 (2021).

We consider the estimation of a Hamiltonian parameter of a set of highly photosensitive samples, which are damaged after a few photons Nabs are absorbed, for a total time T. The samples are modelled as a two mode photonic system, where photons simultaneously acquire information on the unknown parameter and are absorbed at a fixed rate. We show that arbitrarily intense coherent states can obtain information at a rate that scales at most linearly with Nabs and T, whereas quantum states with finite intensity can overcome this bound. We characterise the quantum advantage as a function of Nabs and T, as well as its robustness to imperfections (non-ideal detectors, finite preparation and measurement rates for quantum photonic states). We discuss an implementation in cavity QED, where Fock states are both prepared and measured by coupling atomic ensembles to the cavities. We show that superradiance, arising due to a collective coupling between the cavities and the atoms, can be exploited for improving the speed and efficiency of the measurement.

##### Fractional Chiral Hinge Insulator

A. Hackenbroich, A. Hudomal, N. Schuch, B.A. Bernevig, N. Regnault

Physical Review B 103, L161110 (2021).

We propose and study a wave function describing an interacting three-dimensional fractional chiral hinge insulator (FCHI) constructed by Gutzwiller projection of two noninteracting second-order topological insulators with chiral hinge modes at half filling. We use large-scale variational Monte Carlo computations to characterize the model states via the entanglement entropy and charge-spin fluctuations. We show that the FCHI possesses fractional chiral hinge modes characterized by a central charge c=1 and Luttinger parameter K=1/2, like the edge modes of a Laughlin 1/2 state. The bulk and surface topology is characterized by the topological entanglement entropy (TEE) correction to the area law. While our computations indicate a vanishing bulk TEE, we show that the gapped surfaces host an unconventional two-dimensional topological phase. In a clear departure from the physics of a Laughlin 1/2 state, we find a TEE per surface compatible with (ln√2)/2, half that of a Laughlin 1/2 state. This value cannot be obtained from topological quantum field theory for purely two-dimensional systems. For the sake of completeness, we also investigate the topological degeneracy.

##### Topological Two-Dimensional Floquet Lattice on a Single Superconducting Qubit

D. Malz, A. Smith

Physical Review Letters 126, 163602 (2021).

Current noisy intermediate-scale quantum (NISQ) devices constitute powerful platforms for analog quantum simulation. The exquisite level of control offered by state-of-the-art quantum computers make them especially promising to implement time-dependent Hamiltonians. We implement quasiperiodic driving of a single qubit in the IBM Quantum Experience and thus experimentally realize a temporal version of the half-Bernevig-Hughes-Zhang Chern insulator. Using simple error mitigation, we achieve consistently high fidelities of around 97%. From our data we can infer the presence of a topological transition, thus realizing an earlier proposal of topological frequency conversion by Martin, Refael, and Halperin. Motivated by these results, we theoretically study the many-qubit case, and show that one can implement a wide class of Floquet Hamiltonians, or time-dependent Hamiltonians in general. Our study highlights promises and limitations when studying many-body systems through multifrequency driving of quantum computers.

##### Gaussian continuous tensor network states for simple bosonic field theories

T.D. Karanikolaou, P. Emonts, A. Tilloy

Physical Review Research 3, 023059 (2021).

Tensor networks states allow one to find the low-energy states of local lattice Hamiltonians through variational optimization. Recently, a construction of such states in the continuum was put forward, providing a first step towards the goal of solving quantum field theories (QFTs) variationally. However, the proposed manifold of continuous tensor network states (CTNSs) is difficult to study in full generality, because the expectation values of local observables cannot be computed analytically. In this paper we study a tractable subclass of CTNSs, the Gaussian CTNSs (GCTNSs), and benchmark them on simple quadratic and quartic bosonic QFT Hamiltonians. We show that GCTNSs provide arbitrarily accurate approximations to the ground states of quadratic Hamiltonians and decent estimates for quartic ones at weak coupling. Since they capture the short distance behavior of the theories we consider exactly, GCTNSs even allow one to renormalize away simple divergences variationally. In the end our study makes it plausible that CTNSs are indeed a good manifold to approximate the low-energy states of QFTs.

##### Exact Thermalization Dynamics in the “Rule 54” Quantum Cellular Automaton

K. Klobas, B. Bertini, L. Piroli

Physical Review Letters 126, 160602 (2021).

We study the out-of-equilibrium dynamics of the quantum cellular automaton known as “Rule 54.” For a class of low-entangled initial states, we provide an analytic description of the effect of the global evolution on finite subsystems in terms of simple quantum channels, which gives access to the full thermalization dynamics at the microscopic level. As an example, we provide analytic formulas for the evolution of local observables and Rényi entropies. We show that, in contrast to other known examples of exactly solvable quantum circuits, Rule 54 does not behave as a simple Markovian bath on its own parts, and displays typical nonequilibrium features of interacting integrable many-body quantum systems such as finite relaxation rate and interaction-induced dressing effects. Our study provides a rare example where the full thermalization dynamics can be solved exactly at the microscopic level.

##### Topological Lower Bound on Quantum Chaos by Entanglement Growth

Z.-P. Gong, L. Piroli, J.I. Cirac

Physical Review Letters 126, 160601 (2021).

A fundamental result in modern quantum chaos theory is the Maldacena-Shenker-Stanford upper bound on the growth of out-of-time-order correlators, whose infinite-temperature limit is related to the operator-space entanglement entropy of the evolution operator. Here we show that, for one-dimensional quantum cellular automata (QCA), there exists a lower bound on quantum chaos quantified by such entanglement entropy. This lower bound is equal to twice the index of the QCA, which is a topological invariant that measures the chirality of information flow, and holds for all the Rényi entropies, with its strongest Rényi-∞ version being tight. The rigorous bound rules out the possibility of any sublinear entanglement growth behavior, showing in particular that many-body localization is forbidden for unitary evolutions displaying nonzero index. Since the Rényi entropy is measurable, our findings have direct experimental relevance. Our result is robust against exponential tails which naturally appear in quantum dynamics generated by local Hamiltonians.

##### Field tensor network states

A.E.B. Nielsen, B. Herwerth, J.I. Cirac, G. Sierra

Physical Review B 103, 155130 (2021).

We define a class of tensor network states for spin systems where the individual tensors are functionals of fields. The construction is based on the path-integral representation of correlators of operators in quantum field theory. These tensor network states are infinite-dimensional versions of matrix product states and projected entangled pair states. We find the field tensor that generates the Haldane-Shastry wave function and extend it to two dimensions. We give evidence that the latter underlies the topological chiral state described by the Kalmeyer-Laughlin wave function.

##### Generation of photonic matrix product states with Rydberg atomic arrays

Z.-Y. Wei, D. Malz, A. González-Tudea, J.I. Cirac

Physical Review Research 3, 023021 (2021).

We show how one can deterministically generate photonic matrix product states with high bond and physical dimensions with an atomic array if one has access to a Rydberg-blockade mechanism. We develop both a quantum gate and an optimal control approach to universally control the system and analyze the photon retrieval efficiency of atomic arrays. Comprehensive modeling of the system shows that our scheme is capable of generating a large number of entangled photons. We further develop a multi-port photon emission approach that can efficiently distribute entangled photons into free space in several directions, which can become a useful tool in future quantum networks.

##### Entanglement and complexity of purification in (1+1)-dimensional free conformal field theories

H.A. Camargo, L. Hackl, M.P. Heller, A. Jahn, T. Takayanagi, B. Windt

Physical Review Research 3, 013248 (2021).

Finding pure states in an enlarged Hilbert space that encode the mixed state of a quantum field theory as a partial trace is necessarily a challenging task. Nevertheless, such purifications play the key role in characterizing quantum information-theoretic properties of mixed states via entanglement and complexity of purifications. In this article, we analyze these quantities for two intervals in the vacuum of free bosonic and Ising conformal field theories using the most general Gaussian purifications. We provide a comprehensive comparison with existing results and identify universal properties. We further discuss important subtleties in our setup: the massless limit of the free bosonic theory and the corresponding behavior of the mutual information, as well as the Hilbert space structure under the Jordan-Wigner mapping in the spin chain model of the Ising conformal field theory.

##### Local optimization on pure Gaussian state manifolds

B. Windt, A. Jahn, J. Eisert, L. Hackl

SciPost Physics 10, 066 (2021).

We exploit insights into the geometry of bosonic and fermionic Gaussian states to develop an efficient local optimization algorithm to extremize arbitrary functions on these families of states. The method is based on notions of gradient descent attuned to the local geometry which also allows for the implementation of local constraints. The natural group action of the symplectic and orthogonal group enables us to compute the geometric gradient efficiently. While our parametrization of states is based on covariance matrices and linear complex structures, we provide compact formulas to easily convert from and to other parametrization of Gaussian states, such as wave functions for pure Gaussian states, quasiprobability distributions and Bogoliubov transformations. We review applications ranging from approximating ground states to computing circuit complexity and the entanglement of purification that have both been employed in the context of holography. Finally, we use the presented methods to collect numerical and analytical evidence for the conjecture that Gaussian purifications are sufficient to compute the entanglement of purification of arbitrary mixed Gaussian states.

##### 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.

##### String order parameters for symmetry fractionalization in an enriched toric code

J. Garre-Rubio, M. Iqbal, D.T. Stephen

Physical Review B 103, 125104 (2021).

We study a simple model of symmetry-enriched topological order obtained by decorating a toric code model with lower-dimensional symmetry-protected topological states. We show that the symmetry fractionalization in this model can be characterized by string order parameters, and that these signatures are robust under the effects of external fields and interactions, up to the phase transition point. This extends the recent proposal of Garre-Rubio and Iblisdir [New J. Phys. 21, 113016 (2019)] beyond the setting of fixed-point tensor network states, and solidifies string order parameters as a useful tool to characterize and detect symmetry fractionalization. In addition to this, we observe how the condensation of an anyon that fractionalizes a symmetry forces that symmetry to spontaneously break, and we give a proof of this in the framework of projected entangled pair states. This phenomenon leads to a notable change in the phase diagram of the toric code in parallel magnetic fields

##### Seasonal epidemic spreading on small-world networks: Biennial outbreaks and classical discrete time crystals

D. Malz, A. Pizzi, A. Nunnenkamp, J. Knolle

Physical Review Research 3, 013124 (2021).

We study seasonal epidemic spreading in a susceptible-infected-removed-susceptible model on small-world graphs. We derive a mean-field description that accurately captures the salient features of the model, most notably a phase transition between annual and biennial outbreaks. A numerical scaling analysis exhibits a diverging autocorrelation time in the thermodynamic limit, which confirms the presence of a classical discrete time crystalline phase. We derive the phase diagram of the model both from mean-field theory and from numerics. Our paper demonstrates that small worldness and non-Markovianity can stabilize a classical discrete time crystal, and links recent efforts to understand such dynamical phases of matter to the century-old problem of biennial epidemics.

##### 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.

##### Information Scrambling over Bipartitions: Equilibration, Entropy Production, and Typicality

G. Styliaris, N. Anand, P. Zanardi

Physical Review Letters 126, 030601 (2021).

In recent years, the out-of-time-order correlator (OTOC) has emerged as a diagnostic tool for information scrambling in quantum many-body systems. Here, we present exact analytical results for the OTOC for a typical pair of random local operators supported over two regions of a bipartition. Quite remarkably, we show that this “bipartite OTOC” is equal to the operator entanglement of the evolution, and we determine its interplay with entangling power. Furthermore, we compute long-time averages of the OTOC and reveal their connection with eigenstate entanglement. For Hamiltonian systems, we uncover a hierarchy of constraints over the structure of the spectrum and elucidate how this affects the equilibration value of the OTOC. Finally, we provide operational significance to this bipartite OTOC by unraveling intimate connections with average entropy production and scrambling of information at the level of quantum channels.

##### Fermionic quantum cellular automata and generalized matrix-product unitaries

L. Piroli, A. Turzillo, S.K Shukla, J.I. Cirac

Journal of Statistical Mechanics: Theory and Experiment 013107 (2021).

In this paper, we study matrix-product unitary operators (MPUs) for fermionic one-dimensional chains. In stark contrast to the case of 1D qudit systems, we show that (i) fermionic MPUs (fMPUs) do not necessarily feature a strict causal cone and (ii) not all fermionic quantum cellular automata (QCA) can be represented as fMPUs. We then introduce a natural generalization of the latter, obtained by allowing for an additional operator acting on their auxiliary space. We characterize a family of such generalized MPUs that are locality-preserving, and show that, up to appending inert ancillary fermionic degrees of freedom, any representative of this family is a fermionic QCA (fQCA) and vice versa. Finally, we prove an index theorem for generalized MPUs, recovering the recently derived classification of fQCA in one dimension. As a technical tool for our analysis, we also introduce a graded canonical form for fermionic matrix product states, proving its uniqueness up to similarity transformations.

##### Dominant Fifth-Order Correlations in Doped Quantum Antiferromagnets

A. Bohrdt, Y. Wang, J. Koepsell, M. Kanasz-Nagy, E. Demler, F. Grusdt.

Physical Review Letters 126 (2), 026401 (2021).

Traditionally, one- and two-point correlation functions are used to characterize many-body systems. In strongly correlated quantum materials, such as the doped 2D Fermi-Hubbard system, these may no longer be sufficient, because higher-order correlations are crucial to understanding the character of the many-body system and can be numerically dominant. Experimentally, such higher-order correlations have recently become accessible in ultracold atom systems. Here, we reveal strong non-Gaussian correlations in doped quantum antiferromagnets and show that higher-order correlations dominate over lower-order terms. We study a single mobile hole in the t - J model using the density matrix renormalization group and reveal genuine fifth-order correlations which are directly related to the mobility of the dopant. We contrast our results to predictions using models based on doped quantum spin liquids which feature significantly reduced higher-order correlations. Our predictions can be tested at the lowest currently accessible temperatures in quantum simulators of the 2D Fermi-Hubbard model. Finally, we propose to experimentally study the same fifth-order spin-charge correlations as a function of doping. This will help to reveal the microscopic nature of charge carriers in the most debated regime of the Hubbard model, relevant for understanding high-T-c superconductivity.

##### Time crystallinity and finite-size effects in clean Floquet systems

A. Pizzi, D. Malz, G. De Tomasi, J. Knolle, A. Nunnenkamp

Physical Review B 102 (21), 214207 (2020).

A cornerstone assumption that most literature on discrete time crystals has relied on is that homogeneous Floquet systems generally heat to a featureless infinite temperature state, an expectation that motivated researchers in the field to mostly focus on many-body localized systems. Some works have, however, shown that the standard diagnostics for time crystallinity apply equally well to clean settings without disorder. This fact raises the question whether a homogeneous discrete time crystal is possible in which the originally expected heating is evaded. Studying both a localized and an homogeneous model with short-range interactions, we clarify this issue showing explicitly the key differences between the two cases. On the one hand, our careful scaling analysis confirms that, in the thermodynamic limit and in contrast to localized discrete time crystals, homogeneous systems indeed heat. On the other hand, we show that, thanks to a mechanism reminiscent of quantum scars, finite-size homogeneous systems can still exhibit very crisp signatures of time crystallinity. A subharmonic response can in fact persist over timescales that are much larger than those set by the integrability-breaking terms, with thermalization possibly occurring only at very large system sizes (e.g., of hundreds of spins). Beyond clarifying the emergence of heating in disorder-free systems, our work casts a spotlight on finite-size homogeneous systems as prime candidates for the experimental implementation of nontrivial out-of-equilibrium physics.

##### Gauge redundancy-free formulation of compact QED with dynamical matter for quantum and classical computations

J. Bender, E. Zohar

Physical Review D 102 (11), 114517 (2020).

We introduce a way to express compact quantum electrodynamics with dynamical matter on two- and three-dimensional spatial lattices in a gauge redundancy-free manner while preserving translational invariance. By transforming to a rotating frame, where the matter is decoupled from the gauge constraints, we can express the gauge field operators in terms of dual operators. In two space dimensions, the dual representation is completely free of any local constraints. In three space dimensions, local constraints among the dual operators remain but involve only the gauge field degrees of freedom (and not the matter degrees of freedom). These formulations, which reduce the required Hilbert space dimension, could be useful for both numerical (classical) Hamiltonian computations and quantum simulation or computation.

##### Integrability of one-dimensional Lindbladians from operator-space fragmentation

F.H.L. Essler, L. Piroli

Physical Review E 102 (6), 062210 (2020).

We introduce families of one-dimensional Lindblad equations describing open many-particle quantum systems that are exactly solvable in the following sense: (i) The space of operators splits into exponentially many (in system size) subspaces that are left invariant under the dissipative evolution; (ii) the time evolution of the density matrix on each invariant subspace is described by an integrable Hamiltonian. The prototypical example is the quantum version of the asymmetric simple exclusion process (ASEP) which we analyze in some detail. We show that in each invariant subspace the dynamics is described in terms of an integrable spin-1/2 XXZ Heisenberg chain with either open or twisted boundary conditions. We further demonstrate that Lindbladians featuring integrable operator-space fragmentation can be found in spin chains with arbitrary local physical dimensions.

##### 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.

##### Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

I. Cirac, D. Perez-Garcia, N. Schuch, F. Verstraete

The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. We review how matrix product states and projected entangled pair states describe many-body wavefunctions in terms of local tensors. These tensors express how the entanglement is routed, act as a novel type of non-local order parameter, and we describe how their symmetries are reflections of the global entanglement patterns in the full system. We will discuss how tensor networks enable the construction of real-space renormalization group flows and fixed points, and examine the entanglement structure of states exhibiting topological quantum order. Finally, we provide a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications.

##### Projected Entangled Pair States: Fundamental analytical and numerical limitations

G. Scarpa, A. Molnár, Y. Ge, J. J. García-Ripoll, N. Schuch, D. Pérez-García, S. Iblisdir

Physical Review Letters 125, 210504 (2020).

Matrix product states and projected entangled pair states (PEPS) are powerful analytical and numerical tools to assess quantum many-body systems in one and higher dimensions, respectively. While matrix product states are comprehensively understood, in PEPS fundamental questions, relevant analytically as well as numerically, remain open, such as how to encode symmetries in full generality, or how to stabilize numerical methods using canonical forms. Here, we show that these key problems, as well as a number of related questions, are algorithmically undecidable, that is, they cannot be fully resolved in a systematic way. Our work thereby exposes fundamental limitations to a full and unbiased understanding of quantum many-body systems using PEPS.

##### Zero-temperature phases of the two-dimensional Hubbard-Holstein model: A non-Gaussian exact diagonalization study

Y. Wang, I. Esterlis, T. Shi, J.I. Cirac, E. Demler

Physical Review Research 2 (4), 043258 (2020).

We propose a numerical method which embeds the variational non-Gaussian wave-function approach within exact diagonalization, allowing for efficient treatment of correlated systems with both electron-electron and electron-phonon interactions. Using a generalized polaron transformation, we construct a variational wave function that absorbs entanglement between electrons and phonons into a variational non-Gaussian transformation; exact diagonalization is then used to treat the electronic part of the wave function exactly, thus taking into account high-order correlation effects beyond the Gaussian level. Keeping the full electronic Hilbert space, the complexity is increased only by a polynomial scaling factor relative to the exact diagonalization calculation for pure electrons. As an example, we use this method to study ground-state properties of the two-dimensional Hubbard-Holstein model, providing evidence for the existence of intervening phases between the spin and charge-ordered states. In particular, we find one of the intervening phases has strong charge susceptibility and binding energy, but is distinct from a charge-density-wave ordered state, while the other intervening phase displays superconductivity at weak couplings. This method, as a general framework, can be extended to treat excited states and dynamics, as well as a wide range of systems with both electron-electron and electron-boson interactions.

##### Quantum Cellular Automata, Tensor Networks, and Area Laws

L. Piroli, J.I. Cirac

Physical Review Letters 125 (19), 190402 (2020).

Quantum cellular automata are unitary maps that preserve locality and respect causality. We identify them, in any dimension, with simple tensor networks (projected entangled pair unitary) whose bond dimension does not grow with the system size. As a result, they satisfy an area law for the entanglement entropy they can create. We define other classes of nonunitary maps, the so-called quantum channels, that either respect causality or preserve locality. We show that, whereas the latter obey an area law for the number of quantum correlations they can create, as measured by the quantum mutual information, the former may violate it. We also show that neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size.

##### Variational Approach for Many-Body Systems at Finite Temperature

T. Shi, E. Demler, J.I. Cirac

Physical Review Letters 125 (18), 180602 (2020).

We introduce an equation for density matrices that ensures a monotonic decrease of the free energy and reaches a fixed point at the Gibbs thermal. We build a variational approach for many-body systems that can be applied to a broad class of states, including all bosonic and fermionic Gaussian, as well as their generalizations obtained by unitary transformations, such as polaron transformations in electron-phonon systems. We apply it to the Holstein model on 20 x 20 and 50 x 50 square lattices, and predict phase separation between the superconducting and charge-density wave phases in the strong interaction regime.

##### Real-time dynamics in 2+1D compact QED using complex periodic Gaussian states

J. Bender, P. Emonts, E. Zohar, J.I. Cirac

Physical Review Research 2 (4), 043145 (2020).

We introduce a class of variational states to study ground-state properties and real-time dynamics in (2+1)-dimensional compact QED. These are based on complex Gaussian states which are made periodic to account for the compact nature of the U(1) gauge field. Since the evaluation of expectation values involves infinite sums, we present an approximation scheme for the whole variational manifold. We calculate the ground-state energy density for lattice sizes up to 20×20 and extrapolate to the thermodynamic limit for the whole coupling region. Additionally, we study the string tension both by fitting the potential between two static charges and by fitting the exponential decay of spatial Wilson loops. As the ansatz does not require a truncation in the local Hilbert spaces, we analyze truncation effects which are present in other approaches. The variational states are benchmarked against exact solutions known for the one plaquette case and exact diagonalization results for a Z3 lattice gauge theory. Using the time-dependent variational principle, we study real-time dynamics after various global quenches, e.g., the time evolution of a strongly confined electric field between two charges after a quench to the weak-coupling regime. Up to the points where finite-size effects start to play a role, we observe equilibrating behavior.

##### Quantum simulation of two-dimensional quantum chemistry in optical lattices

J. Argüello-Luengo, A. González-Tudela, T. Shi, P. Zoller, J.I. Cirac

Physical Review Research 2 (4), 042013 (R) (2020).

Benchmarking numerical methods in quantum chemistry is one of the key opportunities that quantum simulators can offer. Here, we propose an analog simulator for discrete two-dimensional quantum chemistry models based on cold atoms in optical lattices. We first analyze how to simulate simple models, such as the discrete versions of H and H+2, using a single fermionic atom. We then show that a single bosonic atom can mediate an effective Coulomb repulsion between two fermions, leading to the analog of molecular hydrogen in two dimensions. We extend this approach to larger systems by introducing as many mediating atoms as fermions, and derive the effective repulsion law. In all cases, we analyze how the continuous limit is approached for increasing optical lattice sizes.

##### Disorder-free localization in a simple U (1) lattice gauge theory

I. Papaefstathiou, A. Smith, J. Knolle

Physical Review B 102 (16), 165132 (2020).

Localization due to the presence of disorder has proven crucial for our current understanding of relaxation in isolated quantum systems. The many-body localized phase constitutes a robust alternative to the thermalization of complex interacting systems, but recently the importance of disorder has been brought into question. A number of disorder-free localization mechanisms have been put forward connected to local symmetries of lattice gauge theories. Here, starting from translationally invariant (1 + 1)-dimensional quantum electrodynamics, we modify the dynamics of the gauge field which allows us to construct a lattice model with a U(1) local gauge symmetry revealing a mechanism of disorder-free localization. We consider two different discretizations of the continuum model resulting in a free-fermion soluble model in one case and an interacting model in the other. We diagnose the localization of our translationally invariant model in the far-from-equilibrium dynamics following a global quantum quench.

##### Geometry of variational methods: dynamics of closed quantum systems

L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, J.I. Cirac

SciPost Physics 9, 048 (2020).

We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: K\"ahler and non-K\"ahler. Traditional variational methods typically require the variational family to be a K\"ahler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-K\"ahler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

##### 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.

##### Realizing a deterministic source of multipartite-entangled photonic qubits

J.-C. Besse, K. Reuer, M. C. Collodo, A. Wulff, L. Wernli, A. Copetudo, D. Malz, P. Magnard, A. Akin, M. Gabureac, G. Norris, J.I. Cirac, A. Wallraff, C. Eichler

Nature Communications 11, 4877 (2020).

Sources of entangled electromagnetic radiation are a cornerstone in quantum information processing and offer unique opportunities for the study of quantum many-body physics in a controlled experimental setting. Generation of multi-mode entangled states of radiation with a large entanglement length, that is neither probabilistic nor restricted to generate specific types of states, remains challenging. Here, we demonstrate the fully deterministic generation of purely photonic entangled states such as the cluster, GHZ, and W state by sequentially emitting microwave photons from a controlled auxiliary system into a waveguide. We tomographically reconstruct the entire quantum many-body state for up to N = 4 photonic modes and infer the quantum state for even larger N from process tomography. We estimate that localizable entanglement persists over a distance of approximately ten photonic qubits.

##### One-particle density matrix of a trapped Lieb–Liniger anyonic gas

S. Scopa, L. Piroli, P. Calabrese

Journal of Statistical Mechanics: Theory and Experiment '093103 (2020).

We provide a thorough characterisation of the zero-temperature one-particle density matrix of trapped interacting anyonic gases in one dimension, exploiting recent advances in the field theory description of spatially inhomogeneous quantum systems. We first revisit homogeneous anyonic gases with point-wise interactions. In the harmonic Luttinger liquid expansion of the one-particle density matrix for finite interaction strength, the non-universal field amplitudes were not yet known. We extract them from the Bethe Ansatz formula for the field form factors, providing an exact asymptotic expansion of this correlation function, thus extending the available results in the Tonks–Girardeau limit. Next, we analyse trapped gases with non-trivial density profiles. By applying recent analytic and numerical techniques for inhomogeneous Luttinger liquids, we provide exact expansions for the one-particle density matrix. We present our results for different confining potentials, highlighting the main differences with respect to bosonic gases.

##### Quantum trimer models and topological SU(3) spin liquids on the kagome lattice

T. Shi, J.I. Cirac, E. Demler

Physical Review Research 2 (3), 033379 (2020).

We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the resonating valence bond wave functions to a trimer and SU(3) setting. We demonstrate that these models carry a Z3 symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization. Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological ninefold degenerate ground space. A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers—it can display either Z3 topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior. Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively. Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.

##### Efficient description of many-body systems with Matrix Product Density Operators

J.G. Jarkovský, A. Molnár, N. Schuch, J.I. Cirac

PRX Quantum 1, 010304 (2020).

Matrix product states form a powerful ansatz for the simulation of a wide range of one-dimensional quantum systems that are in a pure state. Their power stems from the fact that they faithfully approximate states with a low amount of entanglement, the “area law.” However, in order to accurately capture the physics of realistic systems, one generally needs to apply a mixed-state description. In this work, we establish the mixed-state analog of this characterization. We show that one-dimensional mixed states with a low amount of entanglement, quantified by the entanglement of purification, can be efficiently approximated by matrix product density operators.

##### Determinant formula for the field form factor in the anyonic Lieb–Liniger model

L. Piroli, S. Scopa, P. Calabrese

Journal of Physics A 53, 405001 (2020).

We derive an exact formula for the field form factor in the anyonic Lieb–Liniger model, valid for arbitrary values of the interaction c, anyonic parameter κ, and number of particles N. Analogously to the bosonic case, the form factor is expressed in terms of the determinant of an N × N matrix, whose elements are rational functions of the Bethe quasimomenta but explicitly depend on κ. The formula is efficient to evaluate, and provide an essential ingredient for several numerical and analytical calculations. Its derivation consists of three steps. First, we show that the anyonic form factor is equal to the bosonic one between two special off-shell Bethe states, in the standard Lieb–Liniger model. Second, we characterize its analytic properties and provide a set of conditions that uniquely specify it. Finally, we show that our determinant formula satisfies these conditions.

##### Subsystem symmetry enriched topological order in three dimensions

D.T. Stephen, J. Garre-Rubio, A. Dua, D.J. Williamson

Physical Review Research 2 (3), 033331 (2020).

We introduce a model of three-dimensional (3D) topological order enriched by planar subsystem symmetries. The model is constructed starting from the 3D toric code, whose ground state can be viewed as an equal-weight superposition of two-dimensional (2D) membrane coverings. We then decorate those membranes with 2D cluster states possessing symmetry-protected topological order under linelike subsystem symmetries. This endows the decorated model with planar subsystem symmetries under which the looplike excitations of the toric code fractionalize, resulting in an extensive degeneracy per unit length of the excitation. We also show that the value of the topological entanglement entropy is larger than that of the toric code for certain bipartitions due to the subsystem symmetry enrichment. Our model can be obtained by gauging the global symmetry of a short-range entangled model which has symmetry-protected topological order coming from an interplay of global and subsystem symmetries. We study the nontrivial action of the symmetries on boundary of this model, uncovering a mixed boundary anomaly between global and subsystem symmetries. To further study this interplay, we consider gauging several different subgroups of the total symmetry. The resulting network of models, which includes models with fracton topological order, showcases more of the possible types of subsystem symmetry enrichment that can occur in 3D.

##### 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.

##### Computing the renormalization group flow of two-dimensional ϕ4 theory with tensor networks

C. Delcamp, A. Tilloy

Physical Review Research 2 (3), 033278 (2020).

We study the renormalization group flow of ϕ4 theory in two dimensions. Regularizing space into a fine-grained lattice and discretizing the scalar field in a controlled way, we rewrite the partition function of the theory as a tensor network. Combining local truncations and a standard coarse-graining scheme, we obtain the renormalization group flow of the theory as a map in a space of tensors. Aside from qualitative insights, we verify the scaling dimensions at criticality and extrapolate the critical coupling constant fc=λ/μ2 to the continuum to find fcontc=11.0861(90), which favorably compares with alternative methods.

##### Scrambling in random unitary circuits: Exact results

- B. Bertini, L. Piroli

Physical Review B 102 (6), 064305 (2020).

We study the scrambling of quantum information in local random unitary circuits by focusing on the tripartite information proposed by Hosur et al. We provide exact results for the averaged Rényi-2 tripartite information in two cases: (i) the local gates are Haar random and (ii) the local gates are dual-unitary and randomly sampled from a single-site Haar-invariant measure. We show that the latter case defines a one-parameter family of circuits, and prove that for a “maximally chaotic” subset of this family quantum information is scrambled faster than in the Haar-random case. Our approach is based on a standard mapping onto an averaged folded tensor network, that can be studied by means of appropriate recurrence relations. By means of the same method, we also revisit the computation of out-of-time-ordered correlation functions, rederiving known formulas for Haar-random unitary circuits, and presenting an exact result for maximally chaotic random dual-unitary gates.

##### 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.

##### Out-of-horizon correlations following a quench in a relativistic quantum field theory

I. Kukuljan, S. Sotiriadis, G. Takács

Journal of High Energy Physics 7, 224 (2020).

One of the manifestations of relativistic invariance in non-equilibrium quantum field theory is the “horizon effect” a.k.a. light-cone spreading of correlations: starting from an initially short-range correlated state, measurements of two observers at distant space-time points are expected to remain independent until their past light-cones overlap. Surprisingly, we find that in the presence of topological excitations correlations can develop outside of horizon and indeed even between infinitely distant points. We demonstrate this effect for a wide class of global quantum quenches to the sine-Gordon model. We point out that besides the maximum velocity bound implied by relativistic invariance, clustering of initial correlations is required to establish the “horizon effect”. We show that quenches in the sine-Gordon model have an interesting property: despite the fact that the initial states have exponentially decaying correlations and cluster in terms of the bosonic fields, they violate the clustering condition for the soliton fields, which is argued to be related to the non-trivial field topology. The nonlinear dynamics governed by the solitons makes the clustering violation manifest also in correlations of the local bosonic fields after the quench.

##### Nondestructive photon counting in waveguide QED

D. Malz, J.I. Cirac

Physical Review Research 2, 033091 (2020).

Number-resolving single-photon detectors represent a key technology for a host of quantum optics protocols, but despite significant efforts, state-of-the-art devices are limited to few photons. In contrast, state-dependent atom counting in arrays can be done with extremely high fidelity up to hundreds of atoms. We show that in waveguide QED, the problem of photon counting can be reduced to atom counting, by entangling the photonic state with an atomic array in the collective number basis. This is possible as the incoming photons couple to collective atomic states and can be achieved by engineering a second decay channel of an excited atom to a metastable state. Our scheme is robust to disorder and finite Purcell factors, and its fidelity increases with the atom number. Analyzing the state of the re-emitted photons, we further show that if the initial atomic state is a symmetric Dicke state, dissipation engineering can be used to implement a nondestructive photon-number measurement, in which the incident state is scattered into the waveguide unchanged. Our results generalize to related platforms, including superconducting qubits.

##### SU(3)_1 Chiral Spin Liquid on the Square Lattice: A View from Symmetric Projected Entangled Pair States

J.-Y. Chen, S. Capponi, A. Wietek, M. Mambrini, N. Schuch, D. Poilblanc

Physical Review Letters 125 (1), 017201 (2020).

Quantum spin liquids can be faithfully represented and efficiently characterized within the framework of projected entangled pair states (PEPS). Guided by extensive exact diagonalization and density matrix renormalization group calculations, we construct an optimized symmetric PEPS for a SU(3)1 chiral spin liquid on the square lattice. Characteristic features are revealed by the entanglement spectrum (ES) on an infinitely long cylinder. In all three Z3 sectors, the level counting of the linear dispersing modes is in full agreement with SU(3)1 Wess-Zumino-Witten conformal field theory prediction. Special features in the ES are shown to be in correspondence with bulk anyonic correlations, indicating a fine structure in the holographic bulk-edge correspondence. Possible universal properties of topological SU(N)k chiral PEPS are discussed.

##### 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.

##### A random unitary circuit model for black hole evaporation

- L. Piroli, C. Sünderhauf, X.-L. Qi

Journal of High Energy Physics 4, 063 (2020).

Inspired by the Hayden-Preskill protocol for black hole evaporation, we consider the dynamics of a quantum many-body qudit system coupled to an external environment, where the time evolution is driven by the continuous limit of certain 2-local random unitary circuits. We study both cases where the unitaries are chosen with and without a conserved U(1) charge and focus on two aspects of the dynamics. First, we study analytically and numerically the growth of the entanglement entropy of the system, showing that two different time scales appear: one is intrinsic to the internal dynamics (the scrambling time), while the other depends on the system-environment coupling. In the presence of a U(1) conserved charge, we show that the entanglement follows a Page-like behavior in time: it begins to decrease in the middle stage of the “evaporation”, and decreases monotonically afterwards. Second, we study the time needed to retrieve information initially injected in the system from measurements on the environment qudits. Based on explicit numerical computations, we characterize such time both when the retriever has control over the initial configuration or not, showing that different scales appear in the two cases.

##### Fermionic tensor networks for higher order topological insulators from charge pumping

A. Hackenbroich, B.A. Bernevig, N. Schuch, N. Regnault

Physical Review B 101, 115134 (2020).

We apply the charge-pumping argument to fermionic tensor network representations of d-dimensional topological insulators (TIs) to obtain tensor network states (TNSs) for (d+1)-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair state (PEPS) for a Chern insulator starting from a matrix product state (MPS) in d=1 describing pumping in the Su-Schrieffer-Heeger (SSH) model. In extending the argument to second-order TIs, we build a three-dimensional TNS for a chiral hinge TI from a PEPS in d=2 for the obstructed atomic insulator (OAI) of the quadrupole model. The (d+1)-dimensional TNSs obtained in this way have a constant bond dimension inherited from the d-dimensional TNSs in all but one spatial direction, making them candidates for numerical applications. From the d-dimensional models, we identify gapped next-nearest-neighbor Hamiltonians interpolating between the trivial and OAI phases of the fully dimerized SSH and quadrupole models, whose ground states are given by an MPS and a PEPS with a constant bond dimension equal to 2, respectively.

##### Classification of Matrix-Product Unitaries with Symmetries

- Z.P. Gong, C. Sünderhauf, N. Schuch, J.I. Cirac

Physical Review Letters 124 (10), 100402 (2020).

We prove that matrix-product unitaries with on-site unitary symmetries are completely classified by the (chiral) index and the cohomology class of the symmetry group G, provided that we can add trivial and symmetric ancillas with arbitrary on-site representations of G. If the representations in both system and ancillas are fixed to be the same, we can define symmetry-protected indices (SPIs) which quantify the imbalance in the transport associated to each group element and greatly refines the classification. These SPIs are stable against disorder and measurable in interferometric experiments. Our results lead to a systematic construction of two-dimensional Floquet symmetry-protected topological phases beyond the standard classification, and thus shed new light on understanding nonequilibrium phases of quantum matter.

##### 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.

##### Exact dynamics in dual-unitary quantum circuits

L. Piroli, B. Bertini, J.I. Cirac, T. Prosen

Physical Review B 101 (9), 094304 (2020).

We consider the class of dual-unitary quantum circuits in 1+1 dimensions and introduce a notion of “solvable” matrix product states (MPSs), defined by a specific condition which allows us to tackle their time evolution analytically. We provide a classification of the latter, showing that they include certain MPSs of arbitrary bond dimension, and study analytically different aspects of their dynamics. For these initial states, we show that while any subsystem of size ℓ reaches infinite temperature after a time t∝ℓ, irrespective of the presence of conserved quantities, the light cone of two-point correlation functions displays qualitatively different features depending on the ergodicity of the quantum circuit, defined by the behavior of infinite-temperature dynamical correlation functions. Furthermore, we study the entanglement spreading from such solvable initial states, providing a closed formula for the time evolution of the entanglement entropy of a connected block. This generalizes recent results obtained in the context of the self-dual kicked Ising model. By comparison, we also consider a family of nonsolvable initial mixed states depending on one real parameter β, which, as β is varied from zero to infinity, interpolate between the infinite-temperature density matrix and arbitrary initial pure product states. We study analytically their dynamics for small values of β, and highlight the differences from the case of solvable MPSs.

##### Wigner crystals in two-dimensional transition-metal dichalcogenides: Spin physics and readout

J. Knoerzer, M.J.A. Schuetz, G. Giedke, D.S. Wild, K. De Greve, R. Schmidt, M.D. Lukin, J.I. Cirac

Physical Review B 101 (12), 125101 (2020).

Wigner crystals are prime candidates for the realization of regular electron lattices under minimal requirements on external control and electronics. However, several technical challenges have prevented their detailed experimental investigation and applications to date. We propose an implementation of two-dimensional electron lattices for quantum simulation of Ising spin systems based on self-assembled Wigner crystals in transition-metal dichalcogenides. We show that these semiconductors allow for minimally invasive all-optical detection schemes of charge ordering and total spin. For incident light with optimally chosen beam parameters and polarization, we predict a strong dependence of the transmitted and reflected signals on the underlying lattice periodicity, thus revealing the charge order inherent in Wigner crystals. At the same time, the selection rules in transition-metal dichalcogenides provide direct access to the spin degree of freedom via Faraday rotation measurements.

##### Topological Spin Liquids: Robustness under perturbations

M. Iqbal, H. Casademunt, N. Schuch

Physical Review B 101, 115101 (2020).

We study the robustness of the paradigmatic kagome resonating valence bond (RVB) spin liquid and its orthogonal version, the quantum dimer model. The nonorthogonality of singlets in the RVB model and the induced finite length scale not only makes it difficult to analyze, but can also significantly affect its physics, such as how much noise resilience it exhibits. Surprisingly, we find that this is not the case: The amount of perturbations which the RVB spin liquid can tolerate is not affected by the finite correlation length, making the dimer model a viable model for studying RVB physics under perturbations. Remarkably, we find that this is a universal phenomenon protected by symmetries: First, the dominant correlations in the RVB are spinon correlations, making the state robust against doping with visons. Second, reflection symmetry stabilizes the spin liquid against doping with spinons, by forbidding mixing of the initially dominant correlations with those which lead to the breakdown of topological order.

##### Multimode Fock states with large photon number: effective descriptions and applications in quantum metrology

M. Perarnau-Llobet, A. Gonzalez-Tudela, J.I. Cirac

Quantum Science and Technology 5 (2), 025003 (2020).

We develop general tools to characterise and efficiently compute relevant observables of multimode N-photon states generated in nonlinear decays in one-dimensional waveguides. We then consider optical interferometry in a Mach-Zender interferometer where a d-mode photonic state enters in each arm of the interferometer. We derive a simple expression for the quantum Fisher information in terms of the average photon number in each mode, and show that it can be saturated by number-resolved photon measurements that do not distinguish between the different d modes.

##### Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t - J model

C. Hubig, A: Bohrdt, M. Knap, F. Grusdt, J.I. Cirac

Scipost Physics 8 (2), 021 (2020).

Infinite projected entangled pair states (iPEPS) provide a convenient variational description of infinite, translationally-invariant two-dimensional quantum states. However, the simulation of local excitations is not directly possible due to the translationally-invariant ansatz. Furthermore, as iPEPS are either identical or orthogonal, expectation values between different states as required during the evaluation of non-equal-time correlators are ill-defined. Here, we show that by introducing auxiliary states on each site, it becomes possible to simulate both local excitations and evaluate non-equal-time correlators in an iPEPS setting under real-time evolution. We showcase the method by simulating the t - J model after a single hole has been placed in the half-filled antiferromagnetic background and evaluating both return probabilities and spin correlation functions, as accessible in quantum gas microscopes.

##### 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.

##### Expressive power of tensor-network factorizations for probabilistic modeling

I. Glasser, R. Sweke, N. Pancotti, J. Eisert, J.I. Cirac

Advances in Neural Information Processing Systems (NIPS 2019) 32, (2019).

Tensor-network techniques have recently proven useful in machine learning, both as a tool for the formulation of new learning algorithms and for enhancing the mathematical understanding of existing methods. Inspired by these developments, and the natural correspondence between tensor networks and probabilistic graphical models, we provide a rigorous analysis of the expressive power of various tensor-network factorizations of discrete multivariate probability distributions. These factorizations include non-negative tensor-trains/MPS, which are in correspondence with hidden Markov models, and Born machines, which are naturally related to the probabilistic interpretation of quantum circuits. When used to model probability distributions, they exhibit tractable likelihoods and admit efficient learning algorithms. Interestingly, we prove that there exist probability distributions for which there are unbounded separations between the resource requirements of some of these tensor-network factorizations. Of particular interest, using complex instead of real tensors can lead to an arbitrarily large reduction in the number of parameters of the network. Additionally, we introduce locally purified states (LPS), a new factorization inspired by techniques for the simulation of quantum systems, with provably better expressive power than all other representations considered. The ramifications of this result are explored through numerical experiments.

##### Matrix product state algorithms for Gaussian fermionic states

N. Schuch, B. Bauer

Physical Review B 100, 245121 (2019).

While general quantum many-body systems require exponential resources to be simulated on a classical computer, systems of noninteracting fermions can be simulated exactly using polynomially scaling resources. Such systems may be of interest in their own right but also occur as effective models in numerical methods for interacting systems, such as Hartree-Fock, density functional theory, and many others. Often it is desirable to solve systems of many thousand constituent particles, rendering these simulations computationally costly despite their polynomial scaling. We demonstrate how this scaling can be improved by adapting methods based on matrix product states, which have been enormously successful for low-dimensional interacting quantum systems, to the case of free fermions. Compared to the case of interacting systems, our methods achieve an exponential speedup in the entanglement entropy of the state. We demonstrate their use to solve systems of up to one million sites with an effective matrix product state bond dimension of 1015.

##### 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.

##### Quantum chaos in the Brownian SYK model with large finite N : OTOCs and tripartite information

C. Sünderhauf, L. Piroli, X.L. Qi, N. Schuch, J.I. Cirac

Journal of High Energy Physics 38 (2019).

We consider the Brownian SYK model of N interacting Majorana fermions, with random couplings that are taken to vary independently at each time. We study the out-of-time-ordered correlators (OTOCs) of arbitrary observables and the Rényi-2 tripartite information of the unitary evolution operator, which were proposed as diagnostic tools for quantum chaos and scrambling, respectively. We show that their averaged dynamics can be studied as a quench problem at imaginary times in a model of N qudits, where the Hamiltonian displays site-permutational symmetry. By exploiting a description in terms of bosonic collective modes, we show that for the quantities of interest the dynamics takes place in a subspace of the effective Hilbert space whose dimension grows either linearly or quadratically with N , allowing us to perform numerically exact calculations up to N = 106. We analyze in detail the interesting features of the OTOCs, including their dependence on the chosen observables, and of the tripartite information. We observe explicitly the emergence of a scrambling time t∗∼ ln N controlling the onset of both chaotic and scrambling behavior, after which we characterize the exponential decay of the quantities of interest to the corresponding Haar scrambled values.

##### Efficient variational approach to dynamics of a spatially extended bosonic Kondo model

Y. Ashida, T. Shi, R. Schmidt, H.R. Sadeghpour, J.I. Cirac, E. Demler

Physical Review A 100 (4), 043618 (2019).

We develop an efficient variational approach to studying dynamics of a localized quantum spin coupled to a bath of mobile spinful bosons. We use parity symmetry to decouple the impurity spin from the environment via a canonical transformation and reduce the problem to a model of the interacting bosonic bath. We describe coherent time evolution of the latter using bosonic Gaussian states as a variational ansatz. We provide full analytical expressions for equations describing variational time evolution that can be applied to study in- and out-of-equilibrium phenomena in a wide class of quantum impurity problems. In the accompanying paper [Ashida et al., Phys. Rev. Lett. 123, 183001 (2019)], we present a concrete application of this general formalism to the analysis of the Rydberg central spin model, in which the spin-1/2 Rydberg impurity undergoes spin-changing collisions in a dense cloud of two-component ultracold bosons. To illustrate new features arising from orbital motion of the bath atoms, we compare our results to the Monte Carlo study of the model with spatially localized bosons in the bath, in which random positions of the atoms give rise to random couplings of the standard central spin model.

##### Quantum Rydberg Central Spin Model

Y. Ashida, T. Shi, R. Schmidt, H.R. Sadeghpour, J.I. Cirac, E. Demler

Physical Review Letters 123 (8), 183001 (2019).

We consider dynamics of a Rydberg impurity in a cloud of ultracold bosonic atoms in which the Rydberg electron undergoes spin-changing collisions with surrounding atoms. This system realizes a new type of quantum impurity problems that compounds essential features of the Kondo model, the Bose polaron, and the central spin model. To capture the interplay of the Rydberg-electron spin dynamics and the orbital motion of atoms, we employ a new variational method that combines an impurity-decoupling transformation with a Gaussian ansatz for the bath particles. We find several unexpected features of this model that are not present in traditional impurity problems, including interaction-induced renormalization of the absorption spectrum that eludes simple explanations from molecular bound states, and long-lasting oscillations of the Rydberg-electron spin. We discuss generalizations of our analysis to other systems in atomic physics and quantum chemistry, where an electron excitation of high orbital quantum number interacts with a spinful quantum bath.

##### Matrix Product States: Entanglement, Symmetries, and State Transformations

D. Sauerwein, A. Molnar, J.I. Cirac, B. Kraus

Physical Review Letters 123 (7), 170504 (2019).

We analyze entanglement in the family of translationally invariant matrix product states (MPS). We give a criterion to determine when two states can be transformed into each other by local operations with a nonvanishing probability, a central question in entanglement theory. This induces a classification within this family of states, which we explicitly carry out for the simplest, nontrivial MPS. We also characterize all symmetries of translationally invariant MPS, both global and local (inhomogeneous). We illustrate our results with examples of states that are relevant in different physical contexts.

##### Tube algebras, excitations statistics and compactification in gauge models of topological phases

A. Bullivant, C. Delcamp

Journal of High Energy Physics 10, 216 (2019).

We consider lattice Hamiltonian realizations of (d+1)-dimensional Dijkgraaf- Witten theory. In (2+1) d, it is well-known that the Hamiltonian yields point-like excita- tions classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalisation of this strategy that is valid in any dimensions. We then apply this generalisation to derive the algebraic structure of loop-like excitations in (3+1) d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1) d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an R-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a man- ifold that is n-times compactified can be expressed in terms of another model in n-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.

##### Analogue quantum chemistry simulation

J. Argüello-Luengo, A. González-Tudela, T. Shi, P. Zoller, I. Cirac.

Nature 574, 215-218 (2019).

Computing the electronic structure of molecules with high precision is a central challenge in the field of quantum chemistry. Despite the success of approximate methods, tackling this problem exactly with conventional computers remains a formidable task. Several theoretical and experimental attempts have been made to use quantum computers to solve chemistry problems, with early proof-of-principle realizations done digitally. An appealing alternative to the digital approach is analogue quantum simulation, which does not require a scalable quantum computer and has already been successfully applied to solve condensed matter physics problems. However, not all available or planned setups can be used for quantum chemistry problems, because it is not known how to engineer the required Coulomb interactions between them. Here we present an analogue approach to the simulation of quantum chemistry problems that relies on the careful combination of two technologies: ultracold atoms in optical lattices and cavity quantum electrodynamics. In the proposed simulator, fermionic atoms hopping in an optical potential play the role of electrons, additional optical potentials provide the nuclear attraction, and a single-spin excitation in a Mott insulator mediates the electronic Coulomb repulsion with the help of a cavity mode. We determine the operational conditions of the simulator and test it using a simple molecule. Our work opens up the possibility of efficiently computing the electronic structures of molecules with analogue quantum simulation.

##### Entanglement production in the dynamical Casimir effect at parametric resonance

I. Romualdo, L. Hackl, N. YokomizoI. Romualdo, L. Hackl, N. Yokomizo

Physical Review D 100 (6), 065022 (2019).

The particles produced from the vacuum in the dynamical Casimir effect are highly entangled. In order to quantify the correlations generated by the process of vacuum decay induced by moving mirrors, we study the entanglement evolution in the dynamical Casimir effect by computing the time-dependent Rényi and von Neumann entanglement entropy analytically in arbitrary dimensions. We consider the system at parametric resonance, where the effect is enhanced. We find that, in (1+1) dimensions, the entropies grow logarithmically for large times, SA(τ)∼12log(τ), while in higher dimensions (n+1) the growth is linear, SA(t)∼λτ, where λ can be identified with the Lyapunov exponent of a classical instability in the system. In (1+1) dimensions, strong interactions among field modes prevent the parametric resonance from manifesting as a Lyapunov instability, leading to a sublinear entropy growth associated with a constant rate of particle production in the resonant mode. Interestingly, the logarithmic growth comes with a prefactor of 1/2 which cannot occur in time-periodic systems with finitely many degrees of freedom and is thus a special property of bosonic field theories.

##### Gaussian time-dependent variational principle for the Bose-Hubbard model

T. Guaita, L. Hackl, T. Shi, C. Hubig, E. Demler, J. I. Cirac

Physical Review B 100 (9), 094529 (2019).

We systematically extend Bogoliubov theory beyond the mean-field approximation of the Bose-Hubbard model in the superfluid phase. Our approach is based on the time-dependent variational principle applied to the family of all Gaussian states (i.e., Gaussian TDVP). First, we find the best ground-state approximation within our variational class using imaginary time evolution in 1D, 2D, and 3D. We benchmark our results by comparing to Bogoliubov theory and DMRG in 1D. Second, we compute the approximate one- and two-particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP). We find the gapless Goldstone mode, a continuum of two-particle excitations and a doublon mode. We discuss the relation of the gap between Goldstone mode and two-particle continuum to the excitation energy of the Higgs mode. Third, we compute linear response functions for perturbations describing density variation and lattice modulation and discuss their relations to experiment. Our methods can be applied to any perturbations that are linear or quadratic in creation/annihilation operators. Finally, we provide a comprehensive overview how our results are related to well-known methods, such as traditional Bogoliubov theory and random phase approximation.

##### Detecting subsystem symmetry protected topological order via entanglement entropy

D.T. Stephen, H. Dreyer, M. Iqbal, N. Schuch

Physical Review B 100, 115112 (2019).

Subsystem symmetry protected topological (SSPT) order is a type of quantum order that is protected by symmetries acting on lower-dimensional subsystems of the entire system. In this paper, we show how SSPT order can be characterized and detected by a constant correction to the entanglement area law, similar to the topological entanglement entropy. Focusing on the paradigmatic two-dimensional cluster phase as an example, we use tensor network methods to give an analytic argument that almost all states in the phase exhibit the same correction to the area law, such that this correction may be used to reliably detect the SSPT order of the cluster phase. Based on this idea, we formulate a numerical method that uses tensor networks to extract this correction from ground-state wave functions. We use this method to study the fate of the SSPT order of the cluster state under various external fields and interactions, and find that the correction persists unless a phase transition is crossed, or the subsystem symmetry is explicitly broken. Surprisingly, these results uncover that the SSPT order of the cluster state persists beyond the cluster phase, thanks to a new type of subsystem time-reversal symmetry. Finally, we discuss the correction to the area law found in three-dimensional cluster states on different lattices, indicating rich behavior for general subsystem symmetries.

##### Minimal energy cost of entanglement extraction

L. Hackl, R.H. Jonsson

Quantum 3, 165 (2019).

We compute the minimal energy cost for extracting entanglement from the ground state of a bosonic or fermionic quadratic system. Specifically, we find the minimal energy increase in the system resulting from replacing an entangled pair of modes, sharing entanglement entropy ΔS, by a product state, and we show how to construct modes achieving this minimal energy cost. Thus, we obtain a protocol independent lower bound on the extraction of pure state entanglement from quadratic systems. Due to their generality, our results apply to a large range of physical systems, as we discuss with examples.

##### Transport in the sine-Gordon field theory: From generalized hydrodynamics to semiclassics

B. Bertini, L. Piroli, M. Kormos

Physical Review B 100 (3), 035108 (2019).

The semiclassical approach introduced by Sachdev and collaborators proved to be extremely successful in the study of quantum quenches in massive field theories, both in homogeneous and inhomogeneous settings. While conceptually very simple, this method allows one to obtain analytic predictions for several observables when the density of excitations produced by the quench is small. At the same time, a novel generalized hydrodynamic (GHD) approach, which captures exactly many asymptotic features of the integrable dynamics, has recently been introduced. Interestingly, also this theory has a natural interpretation in terms of semiclassical particles and it is then natural to compare the two approaches. This is the objective of this work: we carry out a systematic comparison between the two methods in the prototypical example of the sine-Gordon field theory. In particular, we study the “bipartitioning protocol” where the two halves of a system initially prepared at different temperatures are joined together and then left to evolve unitarily with the same Hamiltonian. We identify two different limits in which the semiclassical predictions are analytically recovered from GHD: a particular nonrelativistic limit and the low-temperature regime. Interestingly, the transport of topological charge becomes subballistic in these cases. Away from these limits we find that the semiclassical predictions are only approximate and, in contrast to the latter, the transport is always ballistic. This statement seems to hold true even for the so-called “hybrid” semiclassical approach, where finite time DMRG simulations are used to describe the evolution in the internal space.

##### Removing staggered fermionic matter in U(N) and SU(N) lattice gauge theories

E. Zohar, J.I. Cirac

Physical Review D 99 (11), 114511 (2019).

Gauge theories, through the local symmetry which is in their core, exhibit many local constraints, that must be taken care of and addressed in any calculation. In the Hamiltonian picture this is phrased through the Gauss laws, which are local constraints that restrict the physical Hilbert space and relate the matter and gauge degrees of freedom. In this work, we present a way that uses all the Gauss laws in lattice gauge theories with staggered fermions for completely removing the matter degrees of freedom, at the cost of locally extending the interaction range, breaking the symmetry and introducing new local constraints, due to the finiteness of the original local matter spaces.

##### 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.

##### 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.

##### Time-dependent study of disordered models with infinite projected entangled pair states

C. Hubig, I. Cirac.

SciPost Physics 6, 031 (2019).

Infinite projected entangled pair states (iPEPS), the tensor network ansatz for two-dimensional systems in the thermodynamic limit, already provide excellent results on ground-state quantities using either imaginary-time evolution or variational optimisation. Here, we show (i) the feasibility of real-time evolution in iPEPS to simulate the dynamics of an infinite system after a global quench and (ii) the application of disorder-averaging to obtain translationally invariant systems in the presence of disorder. To illustrate the approach, we study the short-time dynamics of the square lattice Heisenberg model in the presence of a bi-valued disorder field.

##### Experimentally reducing the quantum measurement back action in work distributions by a collective measurement

K.D. Wu, Y. Yuan, G.-Y. Xiang, C.-F. Li, G.-C. Guo, M. Perarnau-Llobet

Science Advances 5 (3), eaav4944 (2019).

In quantum thermodynamics, the standard approach to estimating work fluctuations in unitary processes is based on two projective measurements, one performed at the beginning of the process and one at the end. The first measurement destroys any initial coherence in the energy basis, thus preventing later interference effects. To decrease this back action, a scheme based on collective measurements has been proposed by Perarnau-Llobet et al. Here, we report its experimental implementation in an optical system. The experiment consists of a deterministic collective measurement on two identically prepared qubit states, encoded in the polarization and path degree of a single photon. The standard two-projective measurement approach is also experimentally realized for comparison. Our results show the potential of collective schemes to decrease the back action of projective measurements, and capture subtle effects arising from quantum coherence.