Projected Entangled Pair States Research Articles

Dual-Isometric Projected Entangled Pair States

Efficient characterization of higher dimensional many-body physical states presents significant challenges. In this Letter, we propose a new class of projected entangled pair states (PEPS) that incorporates two isometric conditions. This new class facilitates the efficient calculation of general local observables and certain two-point correlation functions, which have been previously shown to be intractable for general PEPS, or PEPS with only a single isometric constraint. Despite incorporating two isometric conditions, our class preserves the rich physical structure while enhancing the analytical capabilities. It features a large set of tunable parameters, with only a subleading correction compared to that of general PEPS. Furthermore, we analytically demonstrate that this class can encode universal quantum computation and can represent a transition from topological to trivial order. Published by the American Physical Society 2024

Incommensurate Order with Translationally Invariant Projected Entangled-Pair States: Spiral States and Quantum Spin Liquid on the Anisotropic Triangular Lattice.

Simulating strongly correlated systems with incommensurate order poses significant challenges for traditional finite-size-based approaches. Confining such a phase to a finite-size geometry can induce spurious frustration, with spin spirals in frustrated magnets being a typical example. Here, we introduce an Ansatz based on infinite projected entangled-pair states which overcomes these limitations and enables the direct search for the optimal spiral in the thermodynamic limit, with a computational cost that is independent of the spiral's wavelength. Leveraging this method, we simulate the Heisenberg model on the anisotropic triangular lattice, which interpolates between the square and isotropic triangular lattice limits. Besides accurately reproducing the magnetically ordered phases with arbitrary wavelength, the simulations reveal a quantum spin liquid phase emerging between the Néel and spin spiral phases.

Characterizing Matrix-Product States and Projected Entangled-Pair States Preparable via Measurement and Feedback

Preparing long-range entangled states poses significant challenges for near-term quantum devices. It is known that measurement and feedback (MF) can aid this task by allowing the preparation of certain paradigmatic long-range entangled states with only constant circuit depth. Here, we systematically explore the structure of states that can be prepared using constant-depth local circuits and a single MF round. Using the framework of tensor networks, the preparability under MF translates to tensor symmetries. We detail the structure of matrix-product states (MPSs) and projected entangled-pair states (PEPSs) that can be prepared using MF, revealing the coexistence of Clifford-like properties and magic. In one dimension, we show that states with Abelian-symmetry-protected topological order are a restricted class of MF-preparable states. In two dimensions, we parametrize a subset of states with Abelian topological order that are MF preparable. Finally, we discuss the analogous implementation of operators via MF, providing a structural theorem that connects to the well-known Clifford teleportation. Published by the American Physical Society 2024

A hybrid method integrating Green’s function Monte Carlo and projected entangled pair states

Abstract This paper introduces a hybrid approach combining Green’s function Monte Carlo (GFMC) method with projected entangled pair state (PEPS) ansatz. This hybrid method regards PEPS as a trial state and a guiding wave function in GFMC. By leveraging PEPS’s proficiency in capturing quantum state entanglement and GFMC’s efficient parallel architecture, the hybrid method is well-suited for the accurate and efficient treatment of frustrated quantum spin systems. As a benchmark, we applied this approach to study the frustrated J 1–J 2 Heisenberg model on a square lattice with periodic boundary conditions (PBCs). Compared with other numerical methods, our approach integrating PEPS and GFMC shows competitive accuracy in the performance of ground-state energy. This paper provides systematic and comprehensive discussion of the approach of our previous work [Phys. Rev. B 109 235133 (2024)].

Gauged Gaussian projected entangled pair states: A high dimensional tensor network formulation for lattice gauge theories

Gauge theories form the basis of our understanding of modern physics—ranging from the description of quarks and gluons to effective models in condensed matter physics. In the nonperturbative regime, gauge theories are conventionally treated discretely as lattice gauge theories. The resulting systems are evaluated with path-integral based Monte Carlo methods. These methods, however, can suffer from the sign problem and do not allow for a direct evaluation of real-time dynamics. In this work, we present a unified and comprehensive framework for gauged Gaussian projected entangled pair states (PEPS), a variational ansatz based on tensor networks. We review the construction of Hamiltonian lattice gauge theories, explain their similarities to PEPS, and detail the construction of the ansatz state. The estimation of ground states is based on a variational Monte Carlo procedure with the PEPS as an ansatz state. This sign-problem-free ansatz can be efficiently evaluated in any dimension with arbitrary gauge groups, and can include dynamical fermionic matter, suggesting new options for the simulation of nonperturbative regimes of gauge theories, including quantum chromodynamics. Published by the American Physical Society 2024

Fractional quantum Hall states with variational projected entangled-pair states: A study of the bosonic Harper-Hofstadter model

An important class of model Hamiltonians for investigation of topological phases of matter consists of mobile, interacting particles on a lattice subject to a semiclassical gauge field, as exemplified by the bosonic Harper-Hofstadter model. A unique method for investigations of two-dimensional quantum systems are the infinite projected-entangled pair states, as they avoid spurious finite-size effects that can alter the phase structure. However, due to no-go theorems in related cases, this was often conjectured to be impossible in the past. In this Letter, we show that upon variational optimization, the infinite projected-entangled pair states can be used to this end by identifying fractional Hall states in the bosonic Harper-Hofstadter model. The obtained states are characterized by showing exponential decay of bulk correlations, as dictated by a bulk gap, as well as chiral edge modes via the entanglement spectrum. Published by the American Physical Society 2024

Gauge symmetry of excited states in projected entangled-pair state simulations

While gauge symmetry is a well-established requirement for representing topological orders in projected entangled-pair state (PEPS), its impact on the properties of low-lying excited states remains relatively unexplored. Here we perform PEPS simulations of low-energy dynamics in the Kitaev honeycomb model, which supports fractionalized gauge flux (vison) excitations. We identify gauge symmetry emerging upon optimizing an unbiased PEPS ground state. Using the PEPS adapted local mode approximation, we further classify the low-lying excited states by discerning different vison sectors. Our simulations of spin and spin-dimer dynamical correlations establish close connections with experimental observations. Notably, the selection rule imposed by the locally conserved visons results in nearly flat dispersions in momentum space for excited states belonging to the 2-vison or 4-vison sectors. Published by the American Physical Society 2024

Stacked tree construction for free-fermion projected entangled pair states

The tensor network representation of a state in higher dimensions, say a projected entangled-pair state (PEPS), is typically obtained indirectly through variational optimization or imaginary-time Hamiltonian evolution. Here, we propose a divide-and-conquer approach to directly construct a PEPS representation for free-fermion states admitting descriptions in terms of filling exponentially localized Wannier functions. Our approach relies on first obtaining a tree tensor network description of the state in local subregions. Next, a stacking procedure is used to combine the local trees into a PEPS. Lastly, the local tensors are compressed to obtain a more efficient description. We demonstrate our construction for states in one and two dimensions, including the ground state of an obstructed atomic insulator on the square lattice. Published by the American Physical Society 2024

Generating Function for Projected Entangled-Pair States

Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin-1/2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin-1/2J1-J2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations. Published by the American Physical Society 2024

Tensor network variational optimizations for real-time dynamics: Application to the time-evolution of spin liquids

Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1c=1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.

Optimal Sampling of Dynamical Large Deviations in Two Dimensions via Tensor Networks.

We use projected entangled-pair states (PEPS) to calculate the large deviation statistics of the dynamical activity of the two-dimensional East model, and the two-dimensional symmetric simple exclusion process (SSEP) with open boundaries, in lattices of up to 40×40 sites. We show that at long times both models have phase transitions between active and inactive dynamical phases. For the 2D East model we find that this trajectory transition is of the first order, while for the SSEP we find indications of a second order transition. We then show how the PEPS can be used to implement a trajectory sampling scheme capable of directly accessing rare trajectories. We also discuss how the methods described here can be extended to study rare events at finite times.

Multi-Layered Projected Entangled Pair States for Image Classification

Tensor networks have been recognized as a powerful numerical tool; they are applied in various fields, including physics, computer science, and more. The idea of a tensor network originates from quantum physics as an efficient representation of quantum many-body states and their operations. Matrix product states (MPS) form one of the simplest tensor networks and have been applied to machine learning for image classification. However, MPS has certain limitations when processing two-dimensional images, meaning that it is preferable for an projected entangled pair states (PEPS) tensor network with a similar structure to the image to be introduced into machine learning. PEPS tensor networks are significantly superior to other tensor networks on the image classification task. Based on a PEPS tensor network, this paper constructs a multi-layered PEPS (MLPEPS) tensor network model for image classification. PEPS is used to extract features layer by layer from the image mapped to the Hilbert space, which fully utilizes the correlation between pixels while retaining the global structural information of the image. When performing classification tasks on the Fashion-MNIST dataset, MLPEPS achieves a classification accuracy of 90.44%, exceeding tensor network models such as the original PEPS. On the COVID-19 radiography dataset, MLPEPS has a test set accuracy of 91.63%, which is very close to the results of GoogLeNet. Under the same experimental conditions, the learning ability of MLPEPS is already close to that of existing neural networks while having fewer parameters. MLPEPS can be used to build different network models by modifying the structure, and as such it has great potential in machine learning.

Locality Estimates for Complex Time Evolution in 1D

It is a generalized belief that there are no thermal phase transitions in short range 1D quantum systems. However, the only known case for which this is rigorously proven is for the particular case of finite range translationally invariant interactions. The proof was obtained by Araki in his seminal paper of 1969 as a consequence of pioneering locality estimates for the time-evolution operator that allowed him to prove its analyticity on the whole complex plane, when applied to a local observable. However, as for now there is no mathematical proof of the absence of 1D thermal phase transitions if one allows exponential tails in the interactions. In this work we extend Araki’s result to include exponential (or faster) tails. Our main result is the analyticity of the time-evolution operator applied on a local observable on a suitable strip around the real line. As a consequence we obtain that thermal states in 1D exhibit exponential decay of correlations above a threshold temperature that decays to zero with the exponent of the interaction decay, recovering Araki’s result as a particular case. Our result however still leaves open the possibility of 1D thermal short range phase transitions. We conclude with an application of our result to the spectral gap problem for Projected Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to Cirac et al.

Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states

It is well-known that many topological phase transitions of intrinsic Abelian topological phases are accompanied by condensation and confinement of anyons. However, for non-Abelian topological phases, more intricate phenomena can occur at their phase transitions, because the multiple degenerate degrees of freedom of a non-Abelian anyon can change in different ways after phase transitions. In this paper, we study these new phenomena, including partial condensation, partial deconfinement and especially anyon splitting (a non-Abelian anyon splits into different kinds of the new anyon species) using projected entangled pair states (PEPS). First, we show that anyon splitting can be observed from the topologically degenerate ground states. Next, we construct a set of PEPS describing all possible degrees of freedom of the same non-Abelian anyon. From the overlaps of this set of PEPS of a given non-Abelian anyon with the ground state, we can extract the information of partial condensation. Then, we construct a central object, a matrix defined by the norms and overlaps among the PEPS in that set. The information of partial deconfinement can be extracted from this matrix. In particular, we use it to construct an order parameter which can directly detect anyon splitting. We demonstrate the power of our approach by applying it to a range of non-Abelian topological phase transitions: From $D(S_3)$ quantum double to toric code, from $D(S_3)$ quantum double to $D(Z_3)$ quantum double, from Rep($S_3$) string-net to toric code, and finally from double Ising string-net to toric code.

Real-time dynamics of a critical resonating valence bond spin liquid

Implementation of the hardcore-dimer Hilbert space in cold Rydberg-atom simulators opens a new route of investigating real-time dynamics of dimer liquids under Hamiltonian quench. Here, we consider an initial Resonating Valence Bond (RVB) state on the square lattice realizing a critical Coulomb phase with algebraic and dipolar correlations. Using its representation as a special point of a broad manifold of SU($2$)-symmetric, translationnally invariant, Projected Entangled Pair States (PEPS), we compute its non-equilibrium dynamics upon turning on inter-site Heisenberg interactions. We show that projecting the time-evolution onto the PEPS manifold remains accurate at small time scales. We also find that the state evolves within a PEPS sub-manifold characterized by a U($1$) gauge symmetry, suggesting that the Coulomb phase is stable under such unitary evolution.

Scaling Hypothesis for Projected Entangled-Pair States.

We introduce a new paradigm for scaling simulations with projected entangled-pair states (PEPS) for critical strongly correlated systems, allowing for reliable extrapolations of PEPS data with relatively small bond dimensions D. The key ingredient consists of using the effective correlation length ξ for inducing a collapse of data points, f(D,χ)=f(ξ(D,χ)), for arbitrary values of D and the environment bond dimension χ. As such we circumvent the need for extrapolations in χ and can use many distinct data points for a fixed value of D. Here, we need that the PEPSs have been optimized using a fixed-χ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3D dimer model, the 3D classical Ising model, and the 2D quantum Heisenberg model.

Simulating Chiral Spin Liquids with Projected Entangled-Pair States.

Doubts have been raised on the representation of chiral spin liquids exhibiting topological order in terms of projected entangled pair states (PEPSs). Here, starting from a simple spin-1/2 chiral frustrated Heisenberg model, we show that a faithful representation of the chiral spin liquid phase is in fact possible in terms of a generic PEPS upon variational optimization. We find a perfectly chiral gapless edge mode and a rapid decay of correlation functions at short distances consistent with a bulk gap, concomitant with a gossamer long-range tail originating from a PEPS bulk-edge correspondence. For increasing bond dimension, (i)the rapid decrease of spurious features-SU(2) symmetry breaking and long-range tails in correlations-together with (ii)a faster convergence of the ground state energy as compared to state-of-the-art cylinder matrix-product state simulations involving far more variational parameters, prove the fundamental relevance of the PEPS ansatz for simulating systems with chiral topological order.

Generalized Gibbs ensemble description of real-space entanglement spectra of (2+1) -dimensional chiral topological systems with SU(2) symmetry

We provide a quantitative analysis of the splittings in low-lying numerical entanglement spectra (ES), at given momentum, of a number of quantum states that can be identified, based on "Li-Haldane state-counting", as ground states of (2+1)-dimensional chiral topological phases with global SU(2) symmetry. The ability to account for numerical ES splittings solely within the context of conformal field theory (CFT) is an additional diagnostic of the underlying topological theory, of finer sensitivity than "state-counting". We use the conformal boundary state description of the ES, which can be viewed as a quantum quench. In this language, the ES splittings arise from local conservation laws in the chiral CFT besides the energy, which we view as a Generalized Gibbs Ensemble (GGE). Global SU(2) symmetry imposes strong constraints on the number of such conservation laws, so that only a small number of parameters can be responsible for the splittings. We work out these conservation laws for chiral SU(2) Wess-Zumino-Witten CFTs at levels one and two, and for the latter we notably find that some of the conservation laws take the form of local integrals of operators of fractional dimension, as proposed by Cardy for quantum quenches. We analyze numerical ES from systems with SU(2) symmetry including chiral spin-liquid ground states of local 2D Hamiltonians and two chiral Projected Entangled Pair States (PEPS) tensor networks, which exhibit the "state-counting" of the SU(2)-level-one and -level-two theories. We find that the low-lying ES splittings can be well understood by the lowest of our conservation laws, and we demonstrate the importance of accounting for the fractional conservation laws at level two. Thus the states we consider, including the PEPS, appear chiral also under our more sensitive diagnostic.

Variational methods for contracting projected entangled-pair states

The norms or expectation values of infinite projected entangled-pair states (PEPS) cannot be computed exactly, and approximation algorithms have to be applied. In the last years, many efficient algorithms have been devised -- the corner transfer matrix renormalization group (CTMRG) and variational uniform matrix product state (VUMPS) algorithm are the most common -- but it remains unclear whether they always lead to the same results. In this paper, we identify a subclass of PEPS for which we can reformulate the contraction as a variational problem that is algorithm independent. We use this variational feature to assess and compare the accuracy of CTMRG and VUMPS contractions. Moreover, we devise a new variational contraction scheme, which we can extend to compute general N-point correlation functions.

CPD-Structured Multivariate Polynomial Optimization

We introduce the Tensor-Based Multivariate Optimization (TeMPO) framework for use in nonlinear optimization problems commonly encountered in signal processing, machine learning, and artificial intelligence. Within our framework, we model nonlinear relations by a multivariate polynomial that can be represented by low-rank symmetric tensors (multi-indexed arrays), making a compromise between model generality and efficiency of computation. Put the other way around, our approach both breaks the curse of dimensionality in the system parameters and captures the nonlinear relations with a good accuracy. Moreover, by taking advantage of the symmetric CPD format, we develop an efficient second-order Gauss–Newton algorithm for multivariate polynomial optimization. The presented algorithm has a quadratic per-iteration complexity in the number of optimization variables in the worst case scenario, and a linear per-iteration complexity in practice. We demonstrate the efficiency of our algorithm with some illustrative examples, apply it to the blind deconvolution of constant modulus signals, and the classification problem in supervised learning. We show that TeMPO achieves similar or better accuracy than multilayer perceptrons (MLPs), tensor networks with tensor trains (TT) and projected entangled pair states (PEPS) architectures for the classification of the MNIST and Fashion MNIST datasets while at the same time optimizing for fewer parameters and using less memory. Last but not least, our framework can be interpreted as an advancement of higher-order factorization machines: we introduce an efficient second-order algorithm for higher-order factorization machines.