We introduce a holographic framework for the entanglement Hamiltonian in symmetry-protected topological (SPT) phases with area-law entanglement, whose reduced density matrix (RDM) rho proportional to e-He can be treated as a lower-dimensional mixed state. By replicating rho, we reconstruct the fixed-point SPT wavefunction, establishing an exact correspondence between the bulk strange correlator of the (d+1)-dimensional SPT state and the twisted R & eacute,.nyi-N operator of the d-dimensional RDM. Notably, the RDM exhibits long-range or quasi-long-range order along the replica direction, revealing a universal entanglement feature in SPT phases. As a corollary, we generalized the framework of twisted R & eacute,.nyi-N correlator to thermal states and open quantum systems, providing an alternative formulation of the Lieb-Schultz-Mattis theorem, applicable to both closed and open systems. Finally, we extend our protocol to mixed-state SPT phases and introduce new quantum information metrics-twisted R & eacute,.nyi-N correlators of the surgery operator-to characterize the topology of mixed states.

It is an ongoing quest to realize topologically ordered quantum states on different platforms including condensed matter systems, quantum simulators and digital quantum processors. Unlike conventional states characterized by their local order, these exotic states are characterized by their non-local entanglement. The consequences of topological order can be as profound as they are surprising, ranging from the emergence of fractionalized anyonic excitations to potentially providing a scalable platform for quantum error correction. This deep connection to quantum computing naturally motivates the realization and study of topologically ordered quantum states on quantum processors. However, owing to the non-local nature of these states, their study presents a challenge for near-term quantum devices. This Perspective aims to review the recent progress towards the experimental realization of topologically ordered quantum states, their potential applications and promising directions of future research.

"An important result in the theory of quantum control is the ""universality"" of 2-local unitary gates, i.e., the fact that any global unitary evolution of a system of L qudits can be implemented by composition of 2-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using k-local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of nonuniversality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence."

Quantum machine learning (QML) is an emerging field that investigates the capabilities of quantum computers for learning tasks. While QML models can theoretically offer advantages such as exponential speed-ups, challenges in data loading and the ability to scale to relevant problem sizes have prevented demonstrations of such advantages on practical problems. In particular, the encoding of arbitrary classical data into quantum states usually comes at a high computational cost, either in terms of qubits or gate count. However, real-world data typically exhibits some inherent structure (such as image data) which can be leveraged to load them with a much smaller cost on a quantum computer. This work further develops an efficient algorithm for finding low-depth quantum circuits to load classical image data as quantum states. To evaluate its effectiveness, we conduct systematic studies on the MNIST, Fashion-MNIST, CIFAR-10, and Imagenette datasets. The corresponding circuits for loading the full large-scale datasets are available publicly as PennyLane datasets and can be used by the community for their own benchmarks. We further analyze the performance of various quantum classifiers, such as quantum kernel methods, parameterized quantum circuits, and tensor-network classifiers, and we compare them to convolutional neural networks. In particular, we focus on the performance of the quantum classifiers as we introduce nonlinear functions of the input state, e.g. by letting the circuit parameters depend on the input state.

Motivated by recent experiments, we investigate the spin-12 XXZ model on the triangular lattice with strong Ising anisotropy, combining large-scale numerical simulations and analytical methods to uncover unconventional spin dynamics at T = 0. First, we compute the dynamical spin structure factor using density matrix renormalization group (DMRG) simulations and find excellent agreement with inelastic neutron scattering data on the layered compound K2Co(SeO3)2. The low-energy spectrum reveals a rotonlike minimum at the M point, absent in linear spin-wave theory, accompanied by peak intensity and a broad continuum above it. Near the P point, we observe an approximately linear dispersion with vanishing spectral weight. Second, we compare two analytical frameworks that reproduce the observed features. The first is a hard-core boson approach, which includes (i) an effective staggered boson model at zero magnetic field, (ii) perturbation theory applied to the one-third magnetization plateau, and (iii) a self-consistent mean-field Schwinger boson theory (SBT). The second framework is based on a variational supersolid quantum dimer model (QDM) ansatz, combined with a single-mode approximation. The SBT captures the broad continuum, the M-point minimum, and linear dispersion at P, whereas the QDM reproduces the roton minimum and linear dispersion at finite momentum near P. Remarkably, both the QDM wave function and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photonlike excitations. Together, our comprehensive theoretical and numerical analysis elucidates the microscopic origin of supersolid excitations in the XXZ triangular-lattice model and their proximity to a spin-liquid phase observed experimentally.

Isometric tensor networks in two dimensions enable efficient and accurate study of quantum manybody states, yet the effect of the isometric restriction on the represented quantum states is not fully understood. We address this question in two main contributions. First, we introduce an improved variant of isometric tensor network states (isoTNS) in two dimensions, where the isometric arrows on the columns of the network alternate between pointing upward and downward,. hence the name alternating isometric tensor network states. Second, we introduce a numerical tool-the isometric Gaussian fermionic TNS (isoGfTNS)-that incorporates isometric constraints into the framework of Gaussian fermionic tensor network states. We demonstrate in numerous ways that alternating isoTNSs represent many-body ground states of two-dimensional quantum systems significantly better than the original isoTNSs. First, we show that the entanglement in an isoTNS is mediated along the isometric arrows and that alternating isoTNSs mediate entanglement more efficiently than conventional isoTNSs. Second, alternating isoTNSs correspond to a deeper, and thus more representative, sequential-circuit construction of depth O(Lx <middle dot> Ly) compared to the original isoTNSs of depth O(Lx + Ly). Third, using the Gaussian framework and gradient-based energy minimization, we provide numerical evidence of better bond-dimension scaling and variational energy of alternating isoGfTNSs for ground states of various free-fermionic models, including the Fermi surface, the band insulator, and the px + ipy mean-field superconductor. Finally, benchmarking on the transverse-field Ising model, we demonstrate that an alternating isoTNS provides substantially improved performance and stability relative to the original isoTNS for the ground-state search algorithm in interacting systems.

The simulation of quantum many-body systems, relevant for quantum chemistry and condensed matter physics, is one of the most promising applications of near-term quantum computers before fault-tolerance. However, since the vast majority of quantum computing technologies are built around qubits and discrete gate-based operations, the translation of the physical problem into this framework is a crucial step. This translation will often be device specific, and a suboptimal implementation will be punished by the exponential compounding of errors on real devices. The importance of an efficient mapping is already revealed for models of spinful fermions in two or three dimensions, which naturally arise when the relevant physics relates to electrons. Using the most direct and well-known mapping, the Jordan-Wigner transformation, leads to a nonlocal representation of local degrees of freedom, and necessities efficient decompositions of nonlocal unitary gates into a sequence of hardware accessible local gates. In this paper, we provide a step-by-step recipe for simulating the paradigmatic two-dimensional Fermi-Hubbard model on a quantum computer using only local operations. To provide the ingredients for such a recipe, we briefly review the plethora of different approaches that have emerged recently but focus on the Derby-Klassen compact fermion mapping to make our discussion concrete. We provide a detailed recipe for an end-to-end simulation including embedding on a physical device, preparing initial states such as ground states, simulation of unitary time evolution, and measurement of observables and spectral functions. We explicitly compute the resource requirements for simulating a global quantum quench and conclude by discussing the challenges and future directions for simulating strongly correlated fermionic matter on quantum computers.

Spin ice systems represent a prime example of constrained spin systems and exhibit rich low-energy physics. In this study, we explore how introducing a tunable anisotropic spin coupling to the conventional Ising spin ice Hamiltonian on the breathing pyrochlore lattice affects the ground state properties of the system. Significant changes are observed in the ground state structure, reflected in the spin structure factor and in a reduction of residual entropy at low temperatures. We theoretically uncover a rich phase diagram by varying the anisotropy and demonstrate how this modification reduces the ground state degeneracy across different phases. Numerical simulations reveal that, at sufficiently low temperatures, the system either undergoes a crossover into a constrained spin ice manifold, characterized by an entropy density that drops below the Pauling entropy of conventional spin ice, or a phase transition into a symmetry-broken state, depending on the perturbations. Additionally, we compute the spin structure factors for the anisotropic model and compare these results to analytical predictions from a self-consistent Gaussian approximation, finding good agreement. This work develops the understanding of spin ice in anisotropic limits, which may be experimentally realized by strain, providing, among others, key signatures in entropy and specific heat.

Continuous symmetries lead to universal slow relaxation of correlation functions in quantum many-body systems. In this work, we study how local symmetry-breaking impurities affect the dynamics of these correlation functions using Brownian quantum circuits, which we expect to apply to generic nonintegrable systems with the same symmetries. While explicitly breaking the symmetry is generally expected to lead to eventual restoration of full ergodicity, we find that approximately conserved quantities that survive under such circumstances can still induce slow relaxation. This can be understood using a super-Hamiltonian formulation, where low-lying excitations determine the late-time dynamics and exact ground states correspond to conserved quantities. We show that in one dimension, symmetry-breaking impurities modify diffusive and subdiffusive behaviors associated with U(1) and dipole conservation at late times, e.g., by increasing power-law decay exponents of the decay of autocorrelation functions. This stems from the fact that for these symmetries, impurities are relevant in the renormalization-group sense, e.g., bulk impurities effectively disconnect the system, completely modifying both temporal and spatial correlations. On the other hand, for an impurity that disrupts strong Hilbert space fragmentation, the super-Hamiltonian only acquires an exponentially small gap, leading to prethermal plateaus in autocorrelation functions which extend for times that scale exponentially with the distance to the impurity. Overall, our approach systematically characterizes how symmetry-breaking impurities affect relaxation dynamics in symmetric systems.

We investigate a model of Dirac fermions with mass impurities that open a global topological gap even in the dilute limit. Surprisingly, we find that the chirality of this mass term, i.e., the sign of the Chern number, can be reversed by tuning the magnitude of the single-impurity scattering. Consequently, the disorder induces a phase disconnected from the simple clean limit of the topological insulator, which is achieved via an impurity resonance inducing additional bands with Chern number 2. Thus, we call this impurity induced phase a genuine topological Anderson insulator. In seeming contradiction to the expectation that mass disorder is an irrelevant perturbation to the Dirac semimetal, the tricritical point separating these two Chern insulating phases and a thermal metal phase is located at zero impurity density and connected to the appearance of a zero energy bound state in the continuum that corresponds to a perfectly resonant mass impurity. Our conclusions based on the T-matrix expansion are substantiated by large scale Chebyshevpolynomial Green's function numerics. We discuss possible experimental platforms.

Topologically ordered phases exhibit further complexity in the presence of global symmetries: Their anyonic excitations may exhibit different transformation patterns under these symmetries, leading to a classification in terms of symmetry-enriched topological orders. We develop a generic scheme to study an analogous situation for three-dimensional fracton phases by means of isometric tensor network states (isoTNS) with finite bond dimension, which allow us to tune phase transitions between different symmetry fractionalization patterns. We focus on the X-Cube model, a paradigmatic fracton model hosting two types of excitations: lineons, which are mobile in a single direction only, and fractons that are immobile on their own. By deforming the local tensors of the fixed point ground state, we find a family of exact wavefunctions for which the symmetry fractionalization under an anti-unitary symmetry on both types of excitations is directly visible. These wavefunctions are nonstabilizer states and have nonvanishing correlation lengths. They even exhibit power-law correlations at criticality between two symmetry-enriched topological orders. Furthermore, the isoTNS description allows for the explicit construction of a linear-depth quantum circuit to sequentially realize these exotic 3D states on a quantum processor, including a holographic scheme using only a pair of two-dimensional qubit arrays alongside measurements. Our approach provides a construction to enrich phases with exotic topological or fracton order and to study 3D quantum phase transitions with exact wavefunctions, and offers a tractable route to implement and characterize fracton order on quantum devices.

Out-of-equilibrium phases in many-body systems constitute a new paradigm in quantum matter-they exhibit dynamical properties that may otherwise be forbidden by equilibrium thermodynamics. Among these non-equilibrium phases are periodically driven (Floquet) systems(1-5), which are generically difficult to simulate classically because of their high entanglement. Here we realize a Floquet topologically ordered state theoretically proposed in ref.(6), on an array of superconducting qubits. We image the characteristic dynamics of its chiral edge modes and characterize its emergent anyonic excitations. Devising an interferometric algorithm allows us to introduce and measure a bulk topological invariant to probe the dynamical transmutation of anyons for system sizes up to 58 qubits. Our work demonstrates that quantum processors can provide key insights into the thus-far largely unexplored landscape of highly entangled non-equilibrium phases of matter.

The Quantum Entrepreneurship Lab (QEL) is a one-semester, project-based course at the Technical University of Munich (TUM), designed to bridge the gap between academic research and industrial application in the quantum sector. As part of the Munich Quantum Valley (MQV) ecosystem, the course fosters interdisciplinary collaboration between technical and business students, equipping them with the skills necessary to contribute to or lead in the emerging quantum industry. The QEL curriculum integrates two complementary tracks. First, technical students form teams where they engage in cutting-edge, industry-relevant research topics under academic supervision. Meanwhile business students in a parallel course explore commercialization strategies, risks, and opportunities within the quantum technology landscape. Midway through the semester, a selection of the business students join the technical course to form interdisciplinary teams which assess the feasibility of transforming scientific concepts into viable business solutions. The course culminates in three key deliverables: a publication-style technical report, a white paper analyzing the business potential and financial requirements, and a startup pitch presented to the quantum community at a Demo Day. This work outlines the course structure, objectives, and outcomes, providing a model for other institutions seeking to cultivate a highly skilled, innovation-driven workforce in quantum science and technology.

Within the confined phase of (2 + 1)D lattice gauge theories a roughening transition arises between a weakly confined regime with floppy string excitations and a strongly confined regime with stiff string excitations. In this Letter, we use an infinite Density Matrix Renormalization Group (iDMRG) algorithm to quantitatively characterize the properties of confined strings. To this end, we stabilize the state with a string excitation by 't Hooft loop operators. While for zero gauge-matter coupling we can use bare 't Hooft loop operators to do so, for finite gauge-matter coupling we have to transform them to emergent ones, which we achieve with an adiabatic protocol. By analyzing the scaling of both a novel order parameter and the entanglement entropy, our approach allows us to accurately determine the roughening transition, even at finite gauge-matter coupling.

Lattice gauge theories (LGTs)1, 2, 3-4 can be used to understand a wide range of phenomena, from elementary particle scattering in high-energy physics to effective descriptions of many-body interactions in materials5, 6-7. Studying dynamical properties of emergent phases can be challenging, as it requires solving many-body problems that are generally beyond perturbative limits8, 9-10. Here we investigate the dynamics of local excitations in a Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}}}_{2}$$\end{document} LGT using a two-dimensional lattice of superconducting qubits. We first construct a simple variational circuit that prepares low-energy states that have a large overlap with the ground state,. then we create charge excitations with local gates and simulate their quantum dynamics by means of a discretized time evolution. As the electric field coupling constant is increased, our measurements show signatures of transitioning from deconfined to confined dynamics. For confined excitations, the electric field induces a tension in the string connecting them. Our method allows us to experimentally image string dynamics in a (2+1)D LGT, from which we uncover two distinct regimes inside the confining phase: for weak confinement, the string fluctuates strongly in the transverse direction, whereas for strong confinement, transverse fluctuations are effectively frozen11,12. We also demonstrate a resonance condition at which dynamical string breaking is facilitated. Our LGT implementation on a quantum processor presents a new set of techniques for investigating emergent excitations and string dynamics.

A nonlocal string order parameter detecting topological order and deconfinement has been proposed by Fredenhagen and Marcu (FM). However, due to the lack of exact internal symmetries for lattice models and the nonlinear dependence of the FM string order parameter on ground states, it is a priori not guaranteed that it is a genuine order parameter for topological phase transitions. In this work, we find that the FM string order parameter exhibits universal scaling behavior near critical points of charge condensation transitions, by directly evaluating the FM string order parameter in the infinite string-length limit using infinite Projected Entangled Pair States (iPEPS) for the toric code in a magnetic field. Our results thus demonstrate that the FM string order parameter represents a quantitatively well-behaved order parameter. We find that only in the presence of an emergent 1-form symmetry the corresponding FM string order parameter can faithfully detect topological transitions.

The realization of synthetic gauge fields for charge neutral ultracold atoms and the simulation of quantum Hall physics have witnessed remarkable experimental progress. Here, we establish key signatures of fractional quantum Hall systems in their nonequilibrium quantum dynamics. We show that in the lowest Landau level the system generically relaxes subdiffusively. The slow relaxation is understood from emergent conservation laws of the total charge and the associated dipole moment that arises from the effective Hamiltonian projected onto the lowest Landau level, leading to subdiffusive fracton hydrodynamics. We discuss the prospect of rotating quantum gases as well as ultracold atoms in optical lattices for observing this unconventional relaxation dynamics.

We find that the presence of strong non-Abelian symmetries can lead to highly entangled stationary states even for unital quantum channels. We derive exact expressions for the bipartite logarithmic negativity, Re<acute accent>nyi negativities, and operator space entanglement for stationary states restricted to one symmetric subspace, with focus on the trivial subspace. We prove that these apply to open quantum evolutions whose commutants, characterizing all strongly conserved quantities, correspond to either the universal enveloping algebra of a Lie algebra or the Read-Saleur commutants. The latter provides an example of quantum fragmentation, whose dimension is exponentially large in system size. We find a general upper bound for all these quantities given by the logarithm of the dimension of the commutant on the smaller bipartition of the chain. As Abelian examples, we show that strong U(1) symmetries and classical fragmentation lead to separable stationary states in any symmetric subspace. In contrast, for non-Abelian SU(N) symmetries, both logarithmic and Re<acute accent>nyi negativities scale logarithmically with system size. Finally, we prove that, while Re<acute accent>nyi negativities with n > 2 scale logarithmically with system size, the logarithmic negativity (as well as generalized Re<acute accent>nyi negativities with n < 2) exhibits a volume-law scaling for the Read-Saleur commutants. Our derivations rely on the commutant possessing a Hopf algebra structure in the limit of infinitely large systems and, hence, also apply to finite groups and quantum groups.

We propose an unconventional topological quantum phase transition between a higher-order topological insulator (HOTI) and a featureless Mott insulator, both sharing the same symmetry patterns. Our approach constructs an effective theory for the quantum critical point (QCP) by combining a bosonization technique with the coupled-stripe construction of 1D critical spin ladders. This phase transition is characterized by a critical dipole liquid theory with subsystem U (1) symmetry, where the low-energy modes contain a Bose surface along the k_x,k_y axis. This quantum critical point exhibits fracton dynamics and a breakdown of the area law for entanglement entropy, attributed to the presence of the Bose surface. We numerically validate our findings by measuring the entanglement entropy, topological rank-2 Berry phase, and static structure factor throughout the topological transition, comparing these results with our previous approach derived from the percolation picture. A significant new aspect of our phase transition theory is that the infrared (IR) effective theory is governed by short-wavelength fluctuations, demonstrating a unique UV-IR mixing.

"We explore the relationship between higher-form symmetries and entanglement properties in lattice gauge theories with discrete gauge groups, which can exhibit both topologically ordered phases and higherform symmetry-protected topological (SPT) phases. Our study centers on a generalization of the FradkinShenker model describing Z2 lattice gauge theory with matter, where the Gauss law constraint can be either emergent or exact. The phase diagram includes a topologically ordered deconfined phase and a nontrivial SPT phase protected by a 1-form and a 0-form symmetry, among others. We obtain the following key findings: First, the entanglement properties of the model depend on whether the 1-form symmetries and the Gauss law constraint are exact or emergent. For the emergent Gauss law, the entanglement spectrum (ES) of the nontrivial SPT phase exhibits degeneracies, which are robust at low energies against weak perturbations that explicitly break the exact 1-form symmetry. When the Gauss law and the 1-form symmetry are both exact, the ES degeneracy is extensive. This extensive degeneracy turns out to be fragile and can be removed completely by infinitesimal perturbations that explicitly break the exact 1-form symmetry while keeping the Gauss law exact. Second, we consider the ES in the topologically ordered phase where 1-form symmetries are spontaneously broken. In contrast to the ES of the nontrivial SPT phase, we find that spontaneous higher-form symmetry breaking removes ""half"" of the ES levels, leading to a nondegenerate ES in the topologically ordered phase, in general. Third, we derive a connection between spontaneous higher-form symmetry breaking and the topological entanglement entropy (TEE). Using this relation, we investigate the entanglement entropy that can be distilled in the deconfined phase of the original FradkinShenker model using gauge-invariant measurements. We show that the TEE is robust against the measurement when the 1-form symmetry is emergent but it is fragile when the 1-form symmetry is exact. Our results demonstrate the advantage of higher-form symmetries for understanding entanglement properties of gauge theories."

Efficient preparation of many-body ground states is key to harnessing the power of quantum computers in studying quantum many-body systems. In this work, we propose a simple method to design exact linear-depth parameterized quantum circuits which prepare a family of ground states across topological quantum phase transitions in 2D. We achieve this by constructing ground states represented by isometric tensor networks (isoTNS), which form a subclass of tensor network states that are efficiently preparable. By continuously tuning a parameter in the wavefunction, the many-body ground state undergoes quantum phase transitions, exhibiting distinct 2D quantum phases. We illustrate this by constructing an isoTNS path with bond dimension D = 2 interpolating between distinct symmetry-enriched topological (SET) phases. At the transition point, the wavefunction is related to a gapless point in the classical six-vertex model. Furthermore, the critical wavefunction supports a power-law correlation along one spatial direction while remaining long-range ordered in the other spatial direction. We provide an explicit parametrized local quantum circuit for the path and show that the 2D isoTNS can also be efficiently simulated by a holographic quantum algorithm requiring only an 1D array of qubits.

The relaxation behaviour of isolated quantum systems taken out of equilibrium is among the most intriguing questions in many-body physics1. Quantum systems out of equilibrium typically relax to thermal equilibrium states by scrambling local information and building up entanglement entropy. However, kinetic constraints in the Hamiltonian can lead to a breakdown of this fundamental paradigm owing to a fragmentation of the underlying Hilbert space into dynamically decoupled subsectors in which thermalization can be strongly suppressed2-5. Here we experimentally observe Hilbert space fragmentation in a two-dimensional tilted Bose-Hubbard model. Using quantum gas microscopy, we engineer a wide variety of initial states and find a rich set of manifestations of Hilbert space fragmentation involving bulk states, interfaces and defects, that is, two-, one- and zero-dimensional objects. Specifically, uniform initial states with equal particle number and energy differ strikingly in their relaxation dynamics. Inserting controlled defects on top of a global, non-thermalizing chequerboard state, we observe highly anisotropic, subdimensional dynamics, an immediate signature of their fractonic nature6-9. An interface between localized and thermalizing states in turn shows dynamics depending on its orientation. Our results mark the observation of Hilbert space fragmentation beyond one dimension, as well as the concomitant direct observation of fractons, and pave the way for in-depth studies of microscopic transport phenomena in constrained systems. Using quantum gas microscopy, Hilbert space fragmentation and fractonic excitations are observed in a two-dimensional tilted Bose-Hubbard model.

Machine learning tasks are an exciting application for quantum computers, as it has been proven that they can learn certain problems more efficiently than classical ones. Applying quantum machine learning algorithms to classical data can have many important applications, as qubits allow for dealing with exponentially more data than classical bits. However, preparing the corresponding quantum states usually requires an exponential number of gates and therefore may ruin any potential quantum speedups. Here, we show that classical data with a sufficiently quickly decaying Fourier spectrum after being mapped to a quantum state can be well-approximated by states with a small Schmidt rank (i.e., matrix-product states) and we derive explicit error bounds. These approximated states can, in turn, be prepared on a quantum computer with a linear number of nearest-neighbor two-qubit gates. We confirm our results numerically on a set of 1024 x 1024-pixel images taken from the 'Imagenette' and DIV2K datasets. Additionally, we consider different variational circuit ansa<spacing diaeresis>tze and demonstrate numerically that one-dimensional sequential circuits achieve the same compression quality as more powerful ansa<spacing diaeresis>tze.

TeNPy (short for ‘Tensor Network Python’) is a python library for the simulation of strongly correlated quantum systems with tensor networks. The philosophy of this library is to achieve a balance of readability and usability for new-comers, while at the same time providing powerful algorithms for experts. The focus is on MPS algorithms for 1D and 2D lattices, such as DMRG ground state search, as well as dynamics using TEBD, TDVP, or MPO evolution. This article is a companion to the recent version 1.0 release of TeNPy and gives a brief overview of the package.

TeNPy (short for ‘Tensor Network Python’) is a python library for the simulation of strongly correlated quantum systems with tensor networks. The philosophy of this library is to achieve a balance of readability and usability for new-comers, while at the same time providing powerful algorithms for experts. The focus is on MPS algorithms for 1D and 2D lattices, such as DMRG ground state search, as well as dynamics using TEBD, TDVP, or MPO evolution. This article is a companion to the recent version 1.0 release of TeNPy and gives a brief overview of the package.

We propose the strongly tilted Bose-Hubbard model as a natural platform to explore Hilbert-space fragmentation (HSF) and fracton dynamics in two dimensions in a setup and regime readily accessible in optical lattice experiments. Using a perturbative ansatz, we find HSF when the model is tuned to the resonant limit of on-site interaction and tilted potential. First, we investigate the quench dynamics of this system and observe numerically that the relaxation dynamics strongly depends on the chosen initial state- one of the key signatures of HSF. Second, we identify fractonic excitations with restricted mobility leading to anomalous transport properties. Specifically, we find excitations that show one-dimensional diffusion (z = 1/2) as well as excitations that show subdiffusive behavior in two dimensions (z = 3/4). Using a cellular automaton, we analyze their dynamics and compare it to an effective hydrodynamic description.

Entanglement of mixed quantum states can be quantified using the partial transpose and its corresponding entanglement measure, the logarithmic negativity. Recently, the notion of partial transpose has been extended to systems of anyons, which are exotic quasiparticles whose exchange statistics go beyond the bosonic and fermionic cases. Studying the fundamental properties of this anyonic partial transpose, we first reveal that when applied to the special case of fermionic systems, it can be reduced to the fermionic partial transpose or its twisted variant depending on whether or not a boundary Majorana fermion is present. Focusing on ground state properties, we find that the anyonic partial transpose captures both the correct entanglement scaling for gapless systems, as predicted by conformal field theory, and the phase transition between a topologically trivial and a nontrivial phase. For non-Abelian anyons and the bipartition geometry, we find a rich multiplet structure in the eigenvalues of the partial transpose, the so-called negativity spectrum, and reveal the possibility of defining both a charge- and an imbalance-resolved negativity.

Quantum fluctuations can inhibit long-range ordering in frustrated magnets and potentially lead to quantum spin liquid (QSL) phases. A prime example are gapless QSLs with emergent U(1) gauge fields, which have been understood to be described in terms of quantum electrodynamics in 2+1 dimension (QED3). Despite several promising candidate materials, however, a complicating factor for their realisation is the presence of other degrees of freedom. In particular lattice distortions can act to relieve magnetic frustration, precipitating conventionally ordered states. In this work, we use field-theoretic arguments as well as extensive numerical simulations to show that the U(1) Dirac QSL on the triangular and kagome lattices exhibits a weak-coupling instability due to the coupling of monopoles of the emergent gauge field to lattice distortions, leading to valence-bond solid ordering. This generalises the spin-Peierls instability of one-dimensional quantum critical spin chains to two-dimensional algebraic QSLs. We study static distortions as well as quantum-mechanical phonons. Even in regimes where the QSL is stable, the singular spin-lattice coupling leads to marked temperature-dependent corrections to the phonon spectrum, which provide salient experimental signatures of spin fractionalisation. We discuss the coupling of QSLs to the lattice as a general tool for their discovery and characterisation. Quantum spin liquids can emerge in frustrated magnets where quantum fluctuations prevent long-range order. Seifert et al. show that spin-lattice couplings can relieve magnetic frustration and destabilise 2D gapless quantum spin liquid states with fractionalised excitations, analogous to the 1D spin-Peierls instability.

The advent of near-term digital quantum computers could offer us an exciting opportunity to investigate quantum many-body phenomena beyond that of classical computing. To make the best use of the hardware available, it is paramount that we have methods that accurately simulate Hamiltonian dynamics for limited circuit depths. In this paper, we propose a method to classically optimize unitary brickwall circuits to approximate quantum time evolution operators. Our method is scalable in system size through the use of tensor networks. We demonstrate that, for various three-body Hamiltonians, our approach produces quantum circuits that can outperform trotterization in both their accuracy and the quantum circuit depth needed to implement the dynamics, with the exact details being dependent on the Hamiltonian. We also explain how to choose an optimal time step that minimizes the combined errors of the quantum device and the brickwall circuit approximation.

In the absence of disorder and interactions, fermions move coherently and their associated charge and energy exhibit ballistic spreading, even at finite energy density. In the presence of weak interactions and a finite energy density, fermion-fermion scattering leads to a crossover between early-time ballistic and late-time diffusive transport. The relevant crossover timescales and the transport coefficients are both functions of interaction strength, but the question of determining the precise functional dependence is likely impossible to answer exactly. In this work we develop a numerical method (fDAOE) which is powerful enough to provide an approximate answer to this question, and which is consistent with perturbative arguments in the limit of very weak interactions. Our algorithm, which adapts the existing dissipation-assisted operator evolution (DAOE) to fermions, is applicable to systems of interacting fermions at high temperatures. The algorithm approximates the exact dynamics by systematically discarding information from high n-point functions, and is tailored to capture noninteracting dynamics exactly. Applying our method to a microscopic model of interacting fermions, we numerically determine crossover timescales and diffusion constants for a wide range of interaction strengths. In the limit of weak interaction strength (A), we demonstrate that the crossover from ballistic to diffusive transport happens at a time tD similar to 1/A2 and that the diffusion constant similarly scales as D similar to 1/A2. We confirm that these scalings are consistent with a perturbative Fermi's golden rule calculation, and we provide a heuristic operator-spreading picture for the crossover between ballistic and diffusive transport.

We study the logarithmic entanglement negativity of symmetry-protected topological (SPT) phases and quantum critical points (QCPs) of one-dimensional noninteracting fermions at finite temperatures. In particular, we consider a free fermion model that realizes not only quantum phase transitions between gapped phases but also an exotic topological phase transition between quantum critical states in the form of the fermionic Lifshitz transition. The bipartite entanglement negativity between adjacent fermion blocks reveals the crossover boundary of the quantum critical fan near the QCP between two gapped phases. Along the critical phase boundary between the gapped phases, the sudden decrease in the entanglement negativity signals the fermionic Lifshitz transition responsible for the change in the topological nature of the QCPs. In addition, the tripartite entanglement negativity between spatially separated fermion blocks counts the number of topologically protected boundary modes for both SPT phases and topologically nontrivial QCPs at zero temperature. However, the long-distance entanglement between the boundary modes vanishes at finite temperatures due to the instability of SPTs, the phases themselves.

Recent progress in material design enables the study of correlated, low-temperature phases and associated anomalous transport in two-dimensional kagome systems. Here, we show that unconventional spin transport can arise in such systems even at elevated temperatures due to emergent dynamical constraints. To demonstrate this effect, we consider a strong-coupling limit of an extended Hubbard model on the kagome lattice with a density of 2/3. We numerically investigate the charge and spin transport by a cellular automaton circuit, allowing us to perform simulations on large systems to long times while preserving the essential conservation laws. The charge dynamics reflects the constraints and can be understood by a Gaussian field theory of a scalar height field. Moreover, the system exhibits a hidden spin conservation law with a dynamic sublattice structure, which enables additional slow relaxation pathways for spin excitations. These features can be directly tested by measuring the dynamic spin structure factor with neutron scattering.

We study the dynamical large deviations of the classical stochastic symmetric simple exclusion process (SSEP) by means of numerical matrix product states. We show that for half filling, long-time trajectories with a large enough imbalance between the number hops in even and odd bonds of the lattice belong to distinct symmetryprotected topological (SPT) phases. Using tensor network techniques, we obtain the large deviation (LD) phase diagram in terms of counting fields conjugate to the dynamical activity and the total hop imbalance. We show the existence of high activity trivial and nontrivial SPT phases (classified according to string order parameters) separated by either a critical phase or a critical point. Using the leading eigenstate of the tilted generator, obtained from infinite-system density-matrix renormalization group simulations, we construct a near-optimal dynamics for sampling the LDs, and show that the SPT phases manifest at the level of rare stochastic trajectories. We also show how to extend these results to other filling fractions, and discuss generalizations to asymmetric SEPs.

In the study of the thermalization of closed quantum systems, the role of kinetic constraints on the temporal dynamics and the eventual thermalization is attracting significant interest. Kinetic constraints typically lead to long-lived metastable states depending on initial conditions. We consider a model of interacting hardcore bosons with an additional kinetic constraint that was originally devised to capture glassy dynamics at high densities. As a main result, we demonstrate that the system is highly prone to localization in the presence of uncorrelated disorder. Adding disorder quickly triggers long-lived dynamics as evidenced in the time evolution of density autocorrelations. Moreover, the kinetic constraint favors localization also in the eigenstates, where a finite-size transition to a many-body localized phase occurs for much lower disorder strengths than for the same model without a kinetic constraint. Our work sheds light on the intricate interplay of kinetic constraints and localization and may provide additional control over many-body localized phases in the time domain.

"Repeated projective measurements in unitary circuits can lead to an entanglement phase transition as the measurement rate is tuned. In this work, we consider a different setting in which the projective measurements are replaced by dynamically chosen unitary gates that minimize the entanglement. This can be seen as a one-dimensional unitary circuit game in which two players get to place unitary gates on randomly assigned bonds at different rates: the ""entangler"" applies a random local unitary gate with the aim of generating extensive (volume -law) entanglement. The ""disentangler,"" based on limited knowledge about the state, chooses a unitary gate to reduce the entanglement entropy on the assigned bond with the goal of limiting to only finite (area -law) entanglement. In order to elucidate the resulting entanglement dynamics, we consider three different scenarios: (i) a classical discrete height model, (ii) a Clifford circuit, and (iii) a general U(4) unitary circuit. We find that both the classical and Clifford circuit models exhibit phase transitions as a function of the rate that the disentangler places a gate, which have similar properties that can be understood through a connection to the stochastic Fredkin chain. In contrast, the entangler always wins when using Haar random unitary gates and we observe extensive, volume -law entanglement for all nonzero rates of entangling."

We investigate the phenomenon of Hilbert space fragmentation (HSF) in open quantum systems and find that it can stabilize highly entangled steady states. For concreteness, we consider the Temperley-Lieb model, which exhibits quantum HSF in an entangled basis, and investigate the Lindblad dynamics under two different couplings. First, we couple the system to a dephasing bath that reduces quantum fragmentation to a classical one with the resulting stationary state being separable. We observe that despite vanishing quantum correlations, classical correlations develop due to fluctuations of the remaining conserved quantities, which we show can be captured by a classical stochastic circuit evolution. Second, we use a coupling that preserves the quantum fragmentation structure. We derive a general expression for the steady state, which has a strong coherent memory of the initial state due to the extensive number of noncommuting conserved quantities. We then show that it is highly entangled as quantified by logarithmic negativity.

We present the dynamical spin structure factor of the antiferromagnetic spin-21 J1-J2 Heisenberg model on a triangular lattice obtained from large-scale matrix-product state simulations. The high frustration due to the combination of antiferromagnetic nearest- and next-nearest-neighbor interactions yields a rich phase diagram. We resolve the low-energy excitations both in the 120 degrees ordered phase and in the putative spin-liquid phase at J2/J1 = 0.125. In the ordered phase, we observe an avoided decay of the lowest magnon branch, demonstrating the robustness of this phenomenon in the presence of gapless excitations. Our findings in the spin-liquid phase chime with the field-theoretical predictions for a gapless Dirac spin liquid, in particular the picture of low-lying monopole excitations at the corners of the Brillouin zone. We comment on possible practical difficulties of distinguishing proximate liquid and solid phases based on the dynamical structure factor.

We study measurement-induced symmetry-protected topological (SPT) order in a wide class of quantum random circuit models by combining calculations within the stabilizer formalism with tensor network simulations. We construct a family of quantum random circuits, generating the out-of-equilibrium version of all generalized cluster models, and derive a set of nonlocal string order parameters to distinguish different SPT phases. We apply this framework to investigate a random circuit realization of the XZX cluster model, and use the string order parameter to demonstrate that the phase diagram is stable against extending the class of unitary gates in the circuit, from Clifford gates to Haar unitaries. We then turn to the XZZX generalized cluster model, and demonstrate the coexistence of SPT order and spontaneous symmetry breaking, by relying on string order parameters and a connected correlation function.

Quantum phase transitions between different topologically ordered phases exhibit rich structures and are generically challenging to study in microscopic lattice models. In this paper, we propose a tensor-network solvable model that allows us to tune between different symmetry enriched topological (SET) phases. Concretely, we consider a decorated two-dimensional toric code model for which the ground state can be expressed as a two-dimensional tensor-network state with bond dimension D = 3 and two tunable parameters. We find that the time-reversal (TR) symmetric system exhibits three distinct phases: (i) an SET toric code phase in which anyons transform nontrivially under TR, (ii) a toric code phase in which TR does not fractionalize, and (iii) a topologically trivial phase that is adiabatically connected to a product state. We characterize the different phases using the topological entanglement entropy and a membrane order parameter that distinguishes the two SET phases. Along the phase boundary between the SET toric code phase and the toric code phase, the model has an enhanced U (1) symmetry and the ground state is a quantum critical loop gas wavefunction whose squared norm is equivalent to the partition function of the classical O(2) model. By duality transformations, this tensor-network solvable model can also be used to describe transitions between SET double-semion phases and between Z2 x ZT2 symmetry protected topological phases in two dimensions.

The identification of microscopic systems describing the low-energy properties of correlated materials has been a central goal of spectroscopic measurements. We demonstrate how two-dimensional (2D) nonlinear spectroscopy can be used to distinguish effective spin systems whose linear responses show similar behavior. Motivated by recent experiments on the quasi-1D Ising magnet CoNb2O6, we focus on two proposed systems- the ferromagnetic twisted Kitaev spin chain with bond dependent interactions and the transverse field Ising chain. The dynamical spin structure factor probed in linear response displays similar broad spectra for both systems from their fermionic domain wall excitations. In sharp contrast, the 2D nonlinear spectra of the two systems show clear qualitative differences: those of the twisted Kitaev spin chain contain off-diagonal peaks originating from the bond dependent interactions and transitions between different fermion bands absent in the transverse field Ising chain. We discuss the different signatures of spin fractionalization in integrable and nonintegrable regimes of the systems and their connection to experiments.

Simulating time evolution of generic quantum many-body systems using classical numerical approaches has an exponentially growing cost either with evolution time or with the system size. In this work we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a one-dimensional uniform system in the thermodynamic limit. This algorithm uses a layered uniform sequential quantum circuit as a variational Ansatz to represent infinite translation-invariant quantum states. We show numerically that this Ansatz requires a number of parameters polynomial in the simulation time for a given accuracy. Furthermore, this favorable scaling of the Ansatz is maintained during our variational evolution algorithm. All steps of the hybrid optimization are designed with near-term digital quantum computers in mind. After benchmarking the evolution algorithm on a classical computer, we demonstrate the measurement of observables of this uniform state using a finite number of qubits on a cloud-based quantum processing unit. With more efficient tensor contraction schemes, this algorithm may also offer improvements as a classical numerical algorithm.

"The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an ""orthogonality column"" of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm (iTEBD2) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry."

Quantum convolutional neural networks (QCNNs) have been introduced as classifiers for gapped quantum phases of matter. Here, we propose a model-independent protocol for training QCNNs to discover order parameters that are unchanged under phase-preserving perturbations. We initiate the training sequence with the fixed-point wave functions of the quantum phase and add translation-invariant noise that respects the symmetries of the system to mask the fixed-point structure on short length scales. We illustrate this approach by training the QCNN on phases protected by time-reversal symmetry in one dimension, and test it on several time-reversal symmetric models exhibiting trivial, symmetry-breaking, and symmetryprotected topological order. The QCNN discovers a set of order parameters that identifies all three phases and accurately predicts the location of the phase boundary. The proposed protocol paves the way toward hardware-efficient training of quantum phase classifiers on a programmable quantum processor.

We study higher dimensional models with symmetric correlated hoppings, which generalize a one-dimensional model introduced in the context of dipole-conserving dynamics. We prove rigorously that whenever the local configuration space takes its smallest non-trivial value, these models exhibit localized behavior due to fragmentation, in any dimension. For the same class of models, we then construct a hierarchy of conserved quantities that are power-law localized at the boundary of the system with increasing powers. Combining these with Mazur's bound, we prove that boundary correlations are infinitely long lived, even when the bulk is not localized. We use our results to construct quantum Hamiltonians that exhibit the analogues of strong zero modes in two and higher dimensions.

Tensor networks provide a useful tool to describe low-dimensional complex many-body systems. Finding efficient algorithms to use these methods for finite-temperature simulations in two dimensions is a continuing challenge. Here, we use the class of recently introduced isometric tensor network states, which can also be directly realized with unitary gates on a quantum computer. We utilize a purification ansatz to efficiently represent thermal states of the transverse field Ising model. By performing an imaginary-time evolution starting from infinite temperature, we find that this approach offers a different way with low computational complexity to represent states at finite temperatures.

Two-dimensional systems such as quantum spin liquids or fractional quantum Hall systems exhibit anyonic excitations that possess more general statistics than bosons or fermions. This exotic statistics makes it chal-lenging to solve even a many-body system of non-interacting anyons. We introduce an algorithm that allows to simulate anyonic tight-binding Hamiltonians on two-dimensional lattices. The algorithm is directly derived from the low energy topological quantum field theory and is suited for general Abelian and non-Abelian anyon models. As concrete examples, we apply the algorithm to study the energy level spacing statistics, which reveals level repulsion for free semions, Fibonacci anyons, and Ising anyons. Additionally, we simulate nonequilibrium quench dynamics, where we observe that the density distribution becomes homogeneous for large times-indicating thermalization.

We propose and benchmark a modified time-evolving block decimation algorithm that uses a truncation scheme based on the QR decomposition instead of the singular value decomposition (SVD). The modification reduces the scaling with the dimension of the physical Hilbert space d from d3 down to d2. Moreover, the QR decomposition has a lower computational complexity than the SVD and allows for highly efficient implementations on GPU hardware. In a benchmark simulation of a global quench in a quantum clock model, we observe a speedup of up to three orders of magnitude comparing QR and SVD based updates on an A100 GPU.

We study the nonlinear response of nonintegrable one-dimensional (1D) spin models using infinite matrix-product state techniques. As a benchmark and demonstration of the method, we first calculate the two-dimensional (2D) coherent spectroscopy for the exactly soluble ferromagnetic transverse field Ising model where excitations are freely moving domain walls. We then investigate the distinct signatures of confined bound states by introducing a longitudinal field and observe the emergence of strong nonrephasinglike signals. To interpret the observed phenomena, we use a two-kink approximation to perturbatively compute the 2D spectra. We find good agreement in comparison with the exact results of the infinite matrix-product state method in the strongly confined regime. We discuss the relevance of our results for quasi-1D Ising spin chain materials, such as CoNb2O6.

Intriguingly, quantum many-body systems may defy thermalization even without disorder. One example is so-called fragmented models, where the many-body Hilbert space fragments into dynamically disconnected subspaces that are not determined by the global symmetries of the model. In this Letter we demonstrate that the tilted one-dimensional Fermi-Hubbard model naturally realizes distinct effective Hamiltonians that are expected to support nonergodic behavior due to fragmentation, even at resonances between the tilt energy and the Hubbard on site interaction. We find that the effective description captures the observed dynamics in experimentally accessible parameter ranges of moderate tilt values. Specifically, we observe a pronounced dependence of the relaxation dynamics on the initial doublon fraction, which directly reveals the microscopic processes of the fragmented model. Our results pave the way for future studies of nonergodic behavior in higher dimensions.

The theory of quantum phase transitions separating different phases with distinct symmetry patterns at zero temperature is one of the foundations of modern quantum many-body physics. Here we demonstrate that the existence of a two-dimensional topological phase transition between a higher-order topological insulator (HOTI) and a trivial Mott insulator with the same symmetry eludes this paradigm. We present a theory of this quantum critical point (QCP) driven by the fluctuations and percolation of the domain walls between a HOTI and a trivial Mott insulator region. Due to the spinon zero modes that decorate the rough corners of the domain walls, the fluctuations of the phase boundaries trigger a spinon-dipole hopping term with fracton dynamics. Hence we find that the QCP is characterized by a critical dipole liquid theory with subsystem U(1) symmetry and the breakdown of the area law entanglement entropy which exhibits a logarithmic enhancement: L ln(L). Using the density matrix renormalization group method, we analyze the dipole stiffness together with the structure factor at the QCP, which provides strong evidence of a critical dipole liquid with a Bose surface, UV-IR mixing, and a dispersion relation omega = kxky.

We investigate the computational power of the recently introduced class of isometric tensor network states (isoTNSs), which generalizes the isometric conditions of the canonical form of one-dimensional matrix-product states to tensor networks in higher dimensions. We discuss several technical details regarding the implementation of isoTNSs-based algorithms and compare different disentanglers-which are essential for an efficient handling of isoTNSs. We then revisit the time evolving block decimation for isoTNSs (TEBD2) and explore its power for real-time evolution of two-dimensional (2D) lattice systems. Moreover, we introduce a density matrix renormalization group algorithm for isoTNSs (DMRG2) that allows to variationally find ground states of 2D lattice systems. As a demonstration and benchmark, we compute the dynamical spin structure factor of 2D quantum spin systems for two paradigmatic models: First, we compare our results for the transverse field Ising model on a square lattice with the prediction of the spin-wave theory. Second, we consider the Kitaev model on the honeycomb lattice and compare it to the result from the exact solution.

The finding of physical realizations of topologically ordered states in experimental settings, from condensed matter to artificial quantum systems, has been the main challenge en route to utilizing their unconventional properties. We show how to realize a large class of topologically ordered states and simulate their quasiparticle excitations on a digital quantum computer. To achieve this, we design a set of linear-depth quantum circuits to generate ground states of general string-net models together with unitary open-string operators to simulate the creation and braiding of Abelian and non-Abelian anyons. We show that the Abelian (non-Abelian) unitary string operators can be implemented with a constant- (linear) depth quantum circuit. Our scheme allows us to directly probe characteristic topological properties, including topological entanglement entropy, braiding statistics, and fusion channels of anyons. Moreover, this set of efficiently prepared topologically ordered states has potential applications in the development of fault-tolerant quantum computers.

The advent of noisy-intermediate scale quantum computers has introduced the exciting possibility of achieving quantum speedups in machine learning tasks. These devices, however, are composed of a small number of qubits and can faithfully run only short circuits. This puts many proposed approaches for quantum machine learning beyond currently available devices. We address the problem of compressing classical data into efficient representations on quantum devices. Our proposed methods allow both the required number of qubits and depth of the quantum circuit to be tuned. We achieve this by using a correspondence between matrix-product states and quantum circuits and further propose a hardware-efficient quantum circuit approach, which we benchmark on the Fashion-MNIST dataset. Finally, we demonstrate that a quantum circuit-based classifier can achieve competitive accuracy with current tensor learning methods using only 11 qubits.

The low-energy physics of interacting quantum systems is typically understood through the identification of the relevant quasiparticles or low-energy excitations and their quantum numbers. We present a quantum information framework that goes beyond this to examine the nature of the entanglement in the corresponding quantum states. We argue that the salient features of the quasiparticles, including their quantum numbers, locality, and fractionalization, are reflected in the entanglement spectrum and in the mutual information. We illustrate these ideas in the specific context of the d = 1 transverse field Ising model with an integrability breaking perturbation.

We extend the notions of multipole and subsystem symmetries to more general spatially modulated symmetries. We uncover two instances with exponential and (quasi)periodic modulations and provide simple microscopic models in one, two, and three dimensions. Seeking to understand their effect on the long-time dynamics, we numerically study a stochastic cellular automaton evolution that obeys such symmetries. We prove that, in one dimension, the periodically modulated symmetries lead to a diffusive scaling of correlations modulated by a finite microscopic momentum. In higher dimensions, these symmetries take the form of lines and surfaces of conserved momenta. These give rise to exotic forms of subdiffusive behavior with a rich spatial structure influenced by lattice-scale features. Exponential modulation, on the other hand, can lead to correlations that are infinitely long-lived at the boundary while decaying exponentially in the bulk.

The scaling of the entanglement entropy at a quantum critical point allows us to extract universal properties of the state, e.g., the central charge of a conformal field theory. With the rapid improvement of noisy intermediate-scale quantum (NISQ) devices, these quantum computers present themselves as a powerful tool to study critical many-body systems. We use finite-depth quantum circuits suitable for NISQ devices as a variational ansatz to represent ground states of critical, infinite systems. We find universal finite-depth scaling relations for these circuits and verify them numerically at two different critical points, i.e., the critical Ising model with an additional symmetry-preserving term and the critical XXZ model.

Recently, material realizations of the spin 3/2 Kitaev honeycomb model have been proposed, but the model has not been solved by either analytical or numerical methods. Here the authors report exact results for the spin 3/2 model consistent with numerical simulations, and find gapped and gapless quantum spin liquids. The S=3/2 Kitaev honeycomb model (KHM) is a quantum spin liquid (QSL) state coupled to a static Z(2) gauge field. Employing an SO(6) Majorana representation of spin3/2's, we find an exact representation of the conserved plaquette fluxes in terms of static Z(2) gauge fields akin to the S=1/2 KHM which enables us to treat the remaining interacting matter fermion sector in a parton mean-field theory. We uncover a ground-state phase diagram consisting of gapped and gapless QSLs. Our parton description is in quantitative agreement with numerical simulations, and is furthermore corroborated by the addition of a [001] single ion anisotropy (SIA) which continuously connects the gapless Dirac QSL of our model with that of the S=1/2 KHM. In the presence of a weak [111] SIA, we discuss an emergent chiral QSL within a perturbation theory.

Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain(1,2). Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter(3,4). More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement(5-7). Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi-Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.

Tensor product states have proved extremely powerful for simulating the area-law entangled states of manybody systems, such as the ground states of gapped Hamiltonians in one dimension. The applicability of such methods to the dynamics of many-body systems is less clear: The memory required grows exponentially in time in most cases, quickly becoming unmanageable. New methods reduce the memory required by selectively discarding/dissipating parts of the many-body wave function which are expected to have little effect on the hydrodynamic observables typically of interest: For example, some methods discard fine-grained correlations associated with n-point functions, with n exceeding some cutoff ??????. In this paper, we present a theory for the sizes of backflow corrections, i.e., systematic errors due to discarding this fine-grained information. In particular, we focus on their effect on transport coefficients. Our results suggest that backflow corrections are exponentially suppressed in the size of the cutoff ??????. Moreover, the backflow errors themselves have a hydrodynamical expansion, which we elucidate. We test our predictions against numerical simulations run on random unitary circuits and ergodic spin chains. These results lead to the conjecture that transport coefficients in ergodic diffusive systems can be captured to a given precision e with an amount of memory scaling as exp[O(log(e)2)], significantly better than the naive estimate of memory exp[O(poly(E???1))] required by more brute-force methods.

Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial 'finite-size correction' in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar 'finite size corrections' like the EE which leads to deviations from its characteristic logarithmic scaling.

Simulating quantum many-body dynamics on classical computers is a challenging problem due to the exponential growth of the Hilbert space. Artificial neural networks have recently been introduced as a new tool to approximate quantum many-body states. The variational power of the restricted Boltzmann machine quantum states and different shallow and deep neural autoregressive quantum states to simulate the global quench dynamics of a non-integrable quantum Ising chain is benchmarked. It is found that the number of parameters required to represent the quantum state at a given accuracy increases exponentially in time. The growth rate is only slightly affected by the network architecture over a wide range of different design choices: shallow and deep networks, small and large filter sizes, dilated and normal convolutions, and with and without shortcut connections.

"An interacting system subjected to a strong linear potential can host a many-body localized (MBL) phase when being slightly perturbed. This so-called Wannier-Stark or ""tilted-field"" MBL phase inherits many properties from the well-investigated disordered MBL phase, and provides an alternative route to experimentally engineer interacting localized systems without quenched disorder. Herein, the dynamics of entanglement in a Wannier-Stark MBL system coupled to a dephasing environment is investigated. As an accessible entanglement proxy, the third Renyi negativity R 3 is used, which reduces to the third Renyi entropy in case the system is isolated from the environment. This measure captures the characteristic logarithmic growth of interacting localized phases in the intermediate-time regime, where the effects of the coupling to the environment are not yet dominating the dynamics. Thus, it forms a tool to distinguish Wannier-Stark MBL from noninteracting Wannier-Stark localization up to intermediate time-scales, and to quantify quantum correlations in mixed-state dynamics."

Quantum computers promise to perform computations beyond the reach of modern computers with profound implications for scientific research. Due to remarkable technological advances, small scale devices are now becoming available for use. One of the most apparent applications for such a device is the study of complex many-body quantum systems, where classical computers are unable to deal with the generic exponential complexity of quantum states. Even zero-temperature equilibrium phases of matter and the transitions between them have yet to be fully classified, with topologically protected phases presenting major difficulties. We construct and measure a continuously parametrized family of states crossing a symmetry protected topological phase transition on the IBM Q quantum computers. We present two complementary methods for measuring string order parameters that reveal the transition, and additionally analyze the effects of noise in the device using simple error models. The simulation that we perform is easily scalable and is a practical demonstration of the utility of near-term quantum computers for the study of quantum phases of matter and their transitions.

We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture and applying an artificial dissipation acting on long operators. We represent the observable as a matrix product operator and show that the dissipation leads to a decay of operator entanglement, allowing us to follow the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.

We investigate the rich quantum phase diagram of Wegner's theory of discrete Ising gauge fields interacting with U (1) symmetric single-component fermion matter hopping on a two-dimensional square lattice. In particular limits, the model reduces to (i) pure Z(2) even and odd gauge theories, (ii) free fermions in a static background of deconfined Z(2) gauge fields, and (iii) the kinetic Rokhsar-Kivelson quantum dimer model at a generic dimer filling. We develop a local transformation that maps the lattice gauge theory onto a model of Z(2) gauge-invariant spin 1/2 degrees of freedom. Using the mapping, we perform numerical density matrix renormalization group calculations that corroborate our understanding of the limits identified above. Moreover, in the absence of the magnetic plaquette term, we reveal signatures of topologically ordered Dirac semimetal and staggered Mott insulator phases at half filling. At strong coupling, the lattice gauge theory displays fracton phenomenology with isolated fermions being completely frozen and dimers exhibiting restricted mobility. In that limit, we predict that in the ground state, dimers form compact clusters, whose hopping is suppressed exponentially in their size. We determine the band structure of the smallest clusters numerically using exact diagonalization. The rich phenomenology discussed in this paper can be probed in analog and digital quantum simulators of discrete gauge theories and in Kitaev spin-orbital liquids.

The Bose-Hubbard model subjected to an effective magnetic field hosts a plethora of phases with different topological orders when tuning the chemical potential. Using the density matrix renormalization group method, we identify several gapped phases near the first Mott lobe at strong interactions. They are connected by a particle-hole symmetry to a variety of quantum Hall states stabilized at low fillings. We characterize phases of both particle and hole type and identify signatures compatible with Laughlin, Moore-Read, and bosonic integer quantum Hall states by calculating the quantized Hall conductance and by extracting the topological entanglement entropy. Furthermore, we analyze the entanglement spectrum of Laughlin states of bosonic particles and holes for a range of interaction strengths, as well as the entanglement spectrum of a Moore-Read state. These results further corroborate the existence of topological states at high fillings, close to the first Mott lobe, as hole analogs of the respective low-filling states.

We introduce techniques for analyzing the structure of quantum states of many-body localized (MBL) spin chains by identifying correlation clusters from pairwise correlations. These techniques proceed by interpreting pairwise correlations in the state as a weighted graph, which we analyze using an established graph theoretic clustering algorithm. We validate our approach by studying the eigenstates of a disordered XXZ spin chain across the MBL to ergodic transition, as well as the nonequilibrium dynamics in the MBL phase following a global quantum quench. We successfully reproduce theoretical predictions about the MBL transition obtained from renormalization group schemes. Furthermore, we identify a clear signature of many-body dynamics analogous to the logarithmic growth of entanglement. The techniques that we introduce are computationally inexpensive and, in combination with matrix product state methods, allow for the study of large-scale localized systems. Moreover, the correlation functions we use are directly accessible in a range of experimental settings, including cold atoms.

The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error-correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of -ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.

We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the 1 + 1d case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors. An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as 1/L14. In addition to such new cases, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer-including a complete characterization of symmetry-enriched 1 + 1d Ising CFTs. Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 + 1d CFT without gapped sectors.

Distinguishing the dynamics of an Anderson insulator from a many-body localized (MBL) phase is an experimentally challenging task. In this work we propose a method based on machine learning techniques to analyze experimental snapshot data to separate the two phases. We show how to train three-dimensional convolutional neural networks (CNNs) using space-time Fock-state snapshots, allowing us to obtain dynamic information about the system. We benchmark our method on a paradigmatic model showing MBL (t-V model with quenched disorder), where we obtain a classification accuracy of approximate to 80% between an Anderson insulator and an MBL phase. We underline the importance of providing temporal information to the CNNs and we show that CNNs learn the crucial difference between an Anderson localized and an MBL phase, namely the difference in the propagation of quantum correlations. Particularly, we show that the misclassified MBL samples are characterized by an unusually slow propagation of quantum correlations, and thus the CNNs label them wrongly as Anderson localized. Finally, we apply our method to the case with quasiperiodic potential, known as the Aubry-Andre model (AA model). We find that the CNNs have more difficulties in separating the two phases. We show that these difficulties are due to the fact that the MBL phase of the AA model is characterized by a slower information propagation for numerically accessible system sizes.

Models whose ground states can be written as an exact matrix-product state (MPS) provide valuable insights into phases of matter. While MPS-solvable models are typically studied as isolated points in a phase diagram, they can belong to a connected network of MPS-solvable models, which we call the MPS skeleton. As a case study where we can completely unearth this skeleton, we focus on the one-dimensional BDI class-noninteracting spinless fermions with time-reversal symmetry. This class, labeled by a topological winding number, contains the Kitaev chain and is Jordan-Wigner-dual to various symmetry-breaking and symmetry-protected topological (SPT) spin chains. We show that one can read off from the Hamiltonian whether its ground state is an MPS: defining a polynomial whose coefficients are the Hamiltonian parameters, MPS-solvability corresponds to this polynomial being a perfect square. We provide an explicit construction of the ground state MPS, its bond dimension growing exponentially with the range of the Hamiltonian. This complete characterization of the MPS skeleton in parameter space has three significant consequences: (i) any two topologically distinct phases in this class admit a path of MPS-solvable models between them, including the phase transition which obeys an area law for its entanglement entropy,. (ii) we illustrate that the subset of MPS-solvable models is dense in this class by constructing a sequence of MPS-solvable models which converge to the Kitaev chain (equivalently, the quantum Ising chain in a transverse field),. (iii) a subset of these MPS states can be particularly efficiently processed on a noisy intermediate-scale quantum computer.

Quantum phase transitions are central to our understanding of why matter at very low temperatures can exhibit starkly different properties upon small changes of microscopic parameters. Accurately locating those transitions is challenging experimentally and theoretically. Here, we show that the antithetic strategy of forcing systems out of equilibrium via sudden quenches provides a route to locate quantum phase transitions. Specifically, we show that such transitions imprint distinctive features in the intermediate-time dynamics, and results after equilibration, of local observables in quantum chaotic spin chains. Furthermore, we show that the effective temperature in the expected thermal-like states after equilibration can exhibit minima in the vicinity of the quantum critical points. We discuss how to test our results in experiments with Rydberg atoms and explore nonequilibrium signatures of quantum critical points in models with topological transitions.

The thermalization of isolated quantum many-body systems is deeply related to fundamental questions of quantum information theory. While integrable or many-body localized systems display non-ergodic behavior due to extensively many conserved quantities, recent theoretical studies have identified a rich variety of more exotic phenomena in between these two extreme limits. The tilted one-dimensional Fermi-Hubbard model, which is readily accessible in experiments with ultracold atoms, emerged as an intriguing playground to study non-ergodic behavior in a clean disorder-free system. While non-ergodic behavior was established theoretically in certain limiting cases, there is no complete understanding of the complex thermalization properties of this model. In this work, we experimentally study the relaxation of an initial charge-density wave and find a remarkably long-lived initial-state memory over a wide range of parameters. Our observations are well reproduced by numerical simulations of a clean system. Using analytical calculations we further provide a detailed microscopic understanding of this behavior, which can be attributed to emergent kinetic constraints. It was predicted that complex thermalizing behaviour can arise in many-body systems in the absence of disorder. Here, the authors observe non-ergodic dynamics in a tilted optical lattice that is distinct from previously studied regimes, and propose a microscopic mechanism that is due to emergent kinetic constrains.

Analog quantum simulation has the potential to be an indispensable technique in the investigation of complex quantum systems. In this work, we numerically investigate a one-dimensional, faithful, analog, quantum electronic circuit simulator built out of Josephson junctions for one of the paradigmatic models of an integrable quantum field theory: the quantum sine-Gordon (qSG) model in 1+1 space-time dimensions. We analyze the lattice model using the density matrix renormalization group technique and benchmark our numerical results with existing Bethe ansatz computations. Furthermore, we perform analytical form-factor calculations for the two-point correlation function of vertex operators, which closely agree with our numerical computations. Finally, we compute the entanglement spectrum of the qSG model. We compare our results with those obtained using the integrable lattice-regularization based on the quantum XYZ chain and show that the quantum circuit model is less susceptible to corrections to scaling compared to the XYZ chain. We provide numerical evidence that the parameters required to realize the qSG model are accessible with modern-day superconducting circuit technology, thus providing additional credence towards the viability of the latter platform for simulating strongly interacting quantum field theories. (C) 2021 The Author(s). Published by Elsevier B.V.

In this paper we study the many-body localization (MBL) transition and relate it to the eigenstate structure in the Fock space. Besides the standard entanglement and multifractal probes, we introduce the radial probability distribution of eigenstate coefficients with respect to the Hamming distance in the Fock space and relate the cumulants of this distribution to the properties of the quasilocal integrals of motion in the MBL phase. We demonstrate nonself-averaging property of the many-body fractal dimension D-q and directly relate it to the jump of D-q as well as of the localization length of the integrals of motion at the MBL transition. We provide an example of the continuous many-body transition confirming the above relation via the self-averaging of D-q in the whole range of parameters. Introducing a simple toy model, which hosts ergodic thermal bubbles, we give analytical evidences both in standard probes and in terms of newly introduced radial probability distribution that the MBL transition in the Fock space is consistent with the avalanche mechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBL transition can been seen as a transition between ergodic states to nonergodic extended states and put the upper bound for the disorder scaling for the genuine Anderson localization transition with respect to the noninteracting case.

Tensor networks permit computational and entanglement resources to be concentrated in interesting regions of Hilbert space. Implemented on NISQ machines they allow simulation of quantum systems that are much larger than the computational machine itself. This is achieved by parallelising the quantum simulation. Here, we demonstrate this in the simplest case,. an infinite, translationally invariant quantum spin chain. We provide Cirq and Qiskit code that translates infinite, translationally invariant matrix product state (iMPS) algorithms to finite-depth quantum circuit machines, allowing the representation, optimisation and evolution of arbitrary one-dimensional systems. The illustrative simulated output of these codes for achievable circuit sizes is given.

We re-examine attempts to study the many-body localization transition using measures that are physically natural on the ergodic/quantum chaotic regime of the phase diagram. Using simple scaling arguments and an analysis of various models for which rigorous results are available, we find that these measures can be particularly adversely affected by the strong finite-size effects observed in nearly all numerical studies of many-body localization. This severely impacts their utility in probing the transition and the localized phase. In light of this analysis, we discuss a recent study (?untajs et al., 2020) of the behaviour of the Thouless energy and level repulsion in disordered spin

"We study the late time relaxation dynamics of a pure U(1) lattice gauge theory in the form of a dimer model on a bilayer geometry. To this end, we first develop a proper notion of hydrodynamic transport in such a system by constructing a global conservation law that can be attributed to the presence of topological solitons. The correlation functions of local objects charged under this conservation law can then be used to study the universal properties of the dynamics at late times, applicable to both quantum and classical systems. Performing the time evolution via classically simulable automata circuits unveils a rich phenomenology of the system's nonequilibrium properties: For a large class of relevant initial states, local charges are effectively restricted to move along one-dimensional ""tubes"" within the quasi-two-dimensional system, displaying fracton-like mobility constraints. The timescale on which these tubes are stable diverges with increasing systems size, yielding a novel mechanism for nonergodic behavior in the thermodynamic limit. We further explore the role of geometry by studying the system in a quasi-one-dimensional limit, where the Hilbert space is strongly fragmented due to the emergence of an extensive number of conserved quantities. This provides an instance of a recently introduced concept of ""statistically localized integrals of motion,"" whose universal anomalous hydrodynamics we determine by a mapping to a problem of classical tracer diffusion. We conclude by discussing how our approach might generalize to study transport in other lattice gauge theories."

In this paper, we present a quantum information framework for the entanglement behavior of the low-energy quasiparticle (QP) excitations in various quantum phases in one-dimensional (1D) systems. We first establish an exact correspondence between the correlation matrix and the QP entanglement Hamiltonian for free fermions and find an extended in-gap state in the QP entanglement Hamiltonian as a consequence of the position uncertainty of the QP. A more general understanding of such an in-gap state can be extended to a Kramers theorem for the QP entanglement Hamiltonian, which also applies to strongly interacting systems. Further, we present a set of ubiquitous entanglement spectrum features, dubbed entanglement fragmentation, conditional mutual information, and measurement-induced nonlocal entanglement for QPs in 1D symmetry protected topological phases. Our result thus provides another framework to identify different phases of matter in terms of their QP entanglement.

The current generation of noisy intermediate-scale quantum computers introduces new opportunities to study quantum many-body systems. In this paper, we show that quantum circuits can provide a dramatically more efficient representation than current classical numerics of the quantum states generated under nonequilibrium quantum dynamics. For quantum circuits, we perform both real- and imaginary-time evolution using an optimization algorithm that is feasible on near-term quantum computers. We benchmark the algorithms by finding the ground state and simulating a global quench of the transverse-field Ising model with a longitudinal field on a classical computer. Furthermore, we implement (classically optimized) gates on a quantum processing unit and demonstrate that our algorithm effectively captures real-time evolution.

The presence of global conserved quantities in interacting systems generically leads to diffusive transport at late times. Here, we show that systems conserving the dipole moment of an associated global charge, or even higher-moment generalizations thereof, escape this scenario, displaying subdiffusive decay instead. Modeling the time evolution as cellular automata for specific cases of dipole- and quadrupole conservation, we numerically find distinct anomalous exponents of the late time relaxation. We explain these findings by analytically constructing a general hydrodynamic model that results in a series of exponents depending on the number of conserved moments, yielding an accurate description of the scaling form of charge correlation functions. We analyze the spatial profile of the correlations and discuss potential experimentally relevant signatures of higher-moment conservation.

The combination of strong disorder and interactions in closed quantum systems can lead to many-body localization (MBL). However, this quantum phase is not stable when the system is coupled to a thermal environment. We investigate how MBL is destroyed in systems that are weakly coupled to a dephasive Markovian environment by focusing on their entanglement dynamics. We numerically study the third Renyi negativity R-3, a recently proposed entanglement proxy based on the negativity that captures the unbounded logarithmic growth in the closed case and that can be computed efficiently with tensor networks. We also show that the decay of R-3 follows a stretched exponential law, similarly to the imbalance, with, however, a smaller stretching exponent.

"We discuss how strongly interacting higher-order symmetry protected topological (HOSPT) phases can be characterized from the entanglement perspective: First, we introduce a topological many-body invariant which reveals the noncommutative algebra between a flux operator and C-n rotations. We argue that this invariant denotes the angular momentum carried by the instanton which is closely related to the discrete Wen-Zee response and the fractional corner charge. Second, we define a new entanglement property, dubbed ""higher-order entanglement,"" to scrutinize and differentiate various higher-order topological phases from a hierarchical sequence of the entanglement structure. We support our claims by numerically studying a super-lattice Bose-Hubbard model that exhibits different HOSPT phases."

"Recent work has focused on exploring many-body localization (MBL) in systems without quenched disorder: one such proposal is Stark MBL in which small perturbations to a strong linear potential yield localization. However, as with conventional MBL, it is challenging to experimentally distinguish between noninteracting localization and true MBL. In this paper, we show that several existing experimental probes, designed specifically to differentiate between these scenarios, work similarly in the Stark MBL setting. In particular, we show that a modified spin-echo response shows clear signs of a power-law decay for Stark MBL while quickly saturating for disorder-free Wannier-Stark localization. Furthermore, we observe the characteristic logarithmic-in-time spreading of quantum mutual information in the Stark MBL regime, and an absence of spreading in a noninteracting Stark-localized system. We also show that there are no significant differences in several existing MBL measures for a system consisting of soft-core bosons with repulsive on-site interactions. Lastly, we discuss why curvature or small disorder are needed for an accurate reproduction of MBL phenomenology and how this may be illustrated in experiment. This also connects with recent progress on Hilbert space fragmentation in ""fractonic"" models with a conserved dipole moment, and we suggest this as an auspicious platform for experimental investigations of these phenomena."

Entanglement or modular Hamiltonians play a crucial role in the investigation of correlations in quantum field theories. In particular, in 1 + 1 space-time dimensions, the spectra of entanglement Hamiltonians of conformal field theories (CFTs) for certain geometries are related to the spectra of the physical Hamiltonians of corresponding boundary CFTs. As a result, conformal invariance allows exact computation of the spectra of the entanglement Hamiltonians for these models. In this work, we perform this computation of the spectrum of the entanglement Hamiltonian for the free compactified boson CFT over a finite spatial interval. We compare the analytical results obtained for the continuum theory with numerical simulations of a lattice-regularized model for the CFT using density matrix renormalization group technique. To that end, we use a lattice regularization provided by superconducting quantum electronic circuits, built out of Josephson junctions and capacitors. Up to non-universal effects arising due to the lattice regularization, the numerical results are compatible with the predictions of the exact computations.

We study higher order topology in the presence of strong interactions in a two-dimensional, experimentally accessible superlattice Bose-Hubbard model with alternating hoppings and strong on-site repulsion. We evaluate the phase diagram of the model around half-filling using the density renormalization group ansatz and find two gapped phases separated by a gapless superfluid region. We demonstrate that the gapped states realize two distinct higher order symmetry protected topological phases, which are protected by a combination of charge conservation and C-4 lattice symmetry. The phases are distinguished in terms of a many-body topological invariant and a quantized, experimentally accessible fractional corner charge that is robust against arbitrary, symmetry preserving edge manipulations. We support our claims by numerically studying the full counting statistics of the corner charge, finding a sharp distribution peaked around the quantized values. Our results allow for a direct comparison with experiments and represent a confirmation of theoretically predicted higher order topology in a strongly interacting system. Experimentally, the fractional corner charge can be observed in ultracold atomic settings using state of the art quantum gas microscopy.

Angle-resolved photoemission spectroscopy (ARPES) has revealed peculiar properties of mobile dopants in correlated antiferromagnets (AFMs). But, describing them theoretically, even in simplified toy models, remains a challenge. Here, we study ARPES spectra of a single mobile hole in the t-J model. Recent progress in the microscopic description of mobile dopants allows us to use a geometric decoupling of spin and charge fluctuations at strong couplings, from which we conjecture a one-to-one relation of the one-dopant spectral function and the spectrum of a constituting spinon in the undoped parent AFM. We thoroughly test this hypothesis for a single hole doped into a two-dimensional Heisenberg AFM by comparing our semianalytical predictions to previous quantum Monte Carlo results and our large-scale time-dependent matrix product state calculations of the spectral function. Our conclusion is supported by a microscopic trial wave function describing spinon-chargon bound states, which captures the momentum and t/J dependence of the quasiparticle residue. From our conjecture we speculate that ARPES measurements in the pseudogap phase of cuprates may directly reveal the Dirac-fermion nature of the constituting spinons. Specifically, we demonstrate that our trial wave function provides a microscopic explanation for the sudden drop of spectral weight around the nodal point associated with the formation of Fermi arcs, assuming that additional frustration suppresses long-range AFM ordering. We benchmark our results by studying the crossover from two to one dimension, where spinons and chargons are confined and deconfined, respectively.

Many-body topological invariants, as quantized highly nonlocal correlators of the many-body wave function, are at the heart of the theoretical description of many-body topological quantum phases, including symmetry-protected and symmetry-enriched topological phases. Here, we propose and analyze a universal toolbox of measurement protocols to reveal many-body topological invariants of phases with global symmetries, which can be implemented in state-of-the-art experiments with synthetic quantum systems, such as Rydberg atoms, trapped ions, and superconducting circuits. The protocol is based on extracting the many-body topological invariants from statistical correlations of randomized measurements, implemented with local random unitary operations followed by site-resolved projective measurements. We illustrate the technique and its application in the context of the complete classification of bosonic symmetry-protected topological phases in one dimension, considering in particular the extended Su-Schrieffer-Heeger spin model, as realized with Rydberg tweezer arrays.

"Certain disorder-free Hamiltonians can be nonergodic due to a strong fragmentation of the Hilbert space into disconnected sectors. Here, we characterize such systems by introducing the notion of ""statistically localized integrals of motion"" (SLIOM), whose eigenvalues label the connected components of the Hilbert space. SLIOMs are not spatially localized in the operator sense, but appear localized to subextensive regions when their expectation value is taken in typical states with a finite density of particles. We illustrate this general concept on several Hamiltonians, both with and without dipole conservation. Furthermore, we demonstrate that there exist perturbations which destroy these integrals of motion in the bulk of the system while keeping them on the boundary. This results in statistically localized strong zero modes, leading to infinitely long-lived edge magnetizations along with a thermalizing bulk, constituting the first example of such strong edge modes in a nonintegrable model. We also show that in a particular example, these edge modes lead to the appearance of topological string order in a certain subset of highly excited eigenstates. Some of our suggested models can be realized in Rydberg quantum simulators."

We show that the combination of charge and dipole conservation-characteristic of fracton systems-leads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total S-z) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value-showcasing nonthermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local S-z basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.

Majorana zero modes (MZM-s) occurring at the edges of a one-dimensional (1D), p-wave, spinless superconductor, in the absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temperature. Cooper-pair and MZM-assisted single-electron tunneling, together with finite charging energy of the mesoscopic islands, give rise to a rich phase diagram of this model. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper pairs, or a second gapless phase. In contrast to the LL of Cooper pairs, this second phase is characterized by nonlocal string correlation functions which decay algebraically due to gapless charge-e excitations. The three phases are separated from each other by phase transitions of either Kosterlitz-Thouless or Ising type. Using a Jordan-Wigner transformation, we map the system to a generalized Bose-Hubbard model with two types of hopping and use DMRG to analyze the different phases and the phase transitions.

Recently, a class of tensor networks called isometric tensor network states (isoTNS) was proposed which generalizes the canonical form of matrix product states to tensor networks in higher dimensions. While this ansatz allows for efficient numerical computations, it remained unclear which phases admit an isoTNS representation. In this work, we show that two-dimensional string-net liquids, which represent a wide variety of topological phases including discrete gauge theories, admit an exact isoTNS representation. We further show that the isometric form can be preserved after applying a finite-depth local quantum circuit. Taken together, these results show that long-range entanglement by itself is not an obstruction to isoTNS representation and suggest that all two-dimensional gapped phases with gappable edges admit an isoTNS representation.

Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 21) version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS-which we demonstrate for the 2D transverse field Ising model.

We study the t-V disordered spinless fermionic chain in the strong-coupling regime, t/V -> 0. Strong interactions highly hinder the dynamics of the model, fragmenting its Hilbert space into exponentially many blocks in system size. Macroscopically, these blocks can be characterized by the number of new degrees of freedom, which we refer to as movers. We focus on two limiting cases: blocks with only one mover and ones with a finite density of movers. The former many-particle block can be exactly mapped to a single-particle Anderson model with correlated disorder in one dimension. As a result, these eigenstates are always localized for any finite amount of disorder. The blocks with a finite density of movers, on the other side, show a many-body localized (MBL) transition that is tuned by the disorder strength. Moreover, we provide numerical evidence that its ergodic phase is diffusive at weak disorder. Approaching the MBL transition, we observe subdiffusive dynamics at finite timescales and find indications that this might be only a transient behavior before crossing over to diffusion.

"We study the growth of entanglement in quantum systems with a conserved quantity exhibiting diffusive transport, focusing on how initial inhomogeneities are imprinted on the entropy. We propose a simple effective model, which generalizes the minimal cut picture of Jonay, Huse, and Nahum [arXiv:803.00089] in such a way that the line tension"" of the cut depends on the local entropy density. In the case of noisy dynamics, this is described by the Kardar-Parisi-Zhang (KPZ) equation coupled to a diffusing field. We investigate the resulting dynamics and find that initial inhomogeneities of the conserved charge give rise to features in the entanglement profile, whose width and height both grow in time as alpha root t. In particular, for a domain wall quench, diffusion restricts entanglement growth to be S-VN less than or similar to root t. We find that for charge density wave initial states, these features in the entanglement profile are present even after the charge density has equilibrated. Our conclusions are supported by numerical results on random circuits and deterministic spin chains."

We review different matrix-product-state (MPS) approaches to study the spreading of operators in generic nonintegrable quantum systems. As a common ground to all methods, we quantify this spreading by means of the Frobenius norm of the commutator of a spreading operator with a local operator, which is usually referred to as the out-of-time-order correlation (OTOC) function. We compare two approaches based on matrix-product states in the Schrodinger picture: the time-dependent block decimation (TEBD) and the time-dependent variational principle (TDVP), as well as TEBD based on matrix-product operators directly in the Heisenberg picture. The results of all methods are compared to numerically exact results using Krylov space exact time evolution. We find that for the Schrodinger picture, the TDVP algorithm performs better than the TEBD algorithm. Moreover, the tails of the OTOC are accurately obtained both by TDVP MPS and TEBD MPO. They are in very good agreement with exact results at short times, and appear to be converged in bond dimensions even at longer times. However, the growth and saturation regimes are not well captured by either of the methods.

"We examine the time evolution of the entanglement spectrum of a small subsystem of a nonintegrable spin chain following a quench from a product state. We identify signatures in this entanglement spectrum of the distinct dynamical velocities (related to entanglement and operator spreading) that control thermalization. We show that the onset of level repulsion in the entanglement spectrum occurs on different timescales depending on the ""entanglement energy,"" and that this dependence reflects the shape of the operator front. Level repulsion spreads across the entire entanglement spectrum on a timescale that is parametrically shorter than that for full thermalization of the subsystem. This timescale is also close to when the mutual information between individual spins at the ends of the subsystem reaches its maximum. We provide an analytical understanding of this phenomenon and show supporting numerical data for both random unitary circuits and a microscopic Hamiltonian."

Quantum states of matter-such as solids, magnets and topological phases-typically exhibit collective excitations (for example, phonons, magnons and anyons)(1). These involve the motion of many particles in the system, yet, remarkably, act like a single emergent entity-a quasiparticle. Known to be long lived at the lowest energies, quasiparticles are expected to become unstable when encountering the inevitable continuum of many-particle excited states at high energies, where decay is kinematically allowed. Although this is correct for weak interactions, we show that strong interactions generically stabilize quasiparticles by pushing them out of the continuum. This general mechanism is straightforwardly illustrated in an exactly solvable model. Using state-of-the-art numerics, we find it at work in the spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLHAF). This is surprising given the expectation of magnon decay in this paradigmatic frustrated magnet. Turning to existing experimental data, we identify the detailed phenomenology of avoided decay in the TLHAF material(2) Ba3CoSb2O9, and even in liquid helium(3-8), one of the earliest instances of quasiparticle decay(9). Our work unifies various phenomena above the universal low-energy regime in a comprehensive description. This broadens our window of understanding of many-body excitations, and provides a new perspective for controlling and stabilizing quantum matter in the strongly interacting regime.

The protons in water ice are subject to so-called ice rules resulting in an extensive ground-state degeneracy. We study how an external electric field reduces this ground-state degeneracy in hexagonal water ice I-h within a minimal model. We observe polarization plateaus when the field is aligned along the [001] and [010] directions. In each case, one plateau occurs at intermediate polarization with reduced but still extensive degeneracy. The remaining ground states can be mapped to dimer models on the honeycomb and the square lattice, respectively. Upon tilting the external field, we observe an order-disorder transition of Kasteleyn type into a plateau at saturated polarization and vanishing entropy. This transition is investigated analytically using the Kasteleyn matrix and numerically using a modified directed-loop Monte Carlo simulation. The protons in both cases exhibit algebraically decaying correlations. Moreover, the features of the static structure factor are discussed.

We consider the quench dynamics of a two-dimensional quantum dimer model and determine the role of its kinematic constraints. We interpret the nonequilibrium dynamics in terms of the underlying equilibrium phase transitions consisting of a Berezinskii-Kosterlitz-Thouless (BKT) transition between a columnar ordered valence bond solid (VBS) and a valence bond liquid (VBL), as well as a first-order transition between a staggered VBS and the VBL. We find that quenches from a columnar VBS are ergodic and both order parameters and spatial correlations quickly relax to their thermal equilibrium. By contrast, the staggered side of the first-order transition does not display thermalization on numerically accessible timescales. Based on the model's kinematic constraints, we uncover a mechanism of relaxation that rests on emergent, highly detuned multidefect processes in a staggered background, which gives rise to slow, glassy dynamics at low temperatures even in the thermodynamic limit.

We introduce a method to efficiently study the dynamical properties of many-body localized systems in the regime of strong disorder and weak interactions. Our method reproduces qualitatively and quantitatively the time evolution with a polynomial effort in system size and independent of the desired time scales. We use our method to study quantum information propagation, correlation functions, and temporal fluctuations in one-and two-dimensional many-body localization systems. Moreover, we outline strategies for a further systematic improvement of the accuracy and we point out relations of our method to recent attempts to simulate the time dynamics of quantum many-body systems in classical or artificial neural networks.

We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Renyi entropies. We argue that the latter generically grow sub-ballistically, as proportional to root t, in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Renyi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.