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arXiv:1405.3259 · Physical Review B

Algorithms for finite projected entangled pair states

Michael Lubasch, J. I. Cirac, Mari Carmen Bañuls

Published Aug 25, 2014• 205 citations
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Abstract

Projected entangled pair states (PEPS) are a promising ansatz for the study of strongly correlated quantum many-body systems in two dimensions. However, due to their high computational cost, developing and improving PEPS algorithms is necessary to make the ansatz widely usable in practice. Here we analyze several algorithmic aspects of the method. On the one hand, we quantify the connection between the correlation length of the PEPS and the accuracy of its approximate contraction and discuss how purifications can be used in the latter. On the other hand, we present algorithmic improvements for the update of the tensor that introduce drastic gains in the numerical conditioning and the efficiency of the algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with the Heisenberg and quantum Ising models on lattices of up to $21\ifmmode\times\else\texttimes\fi{}21$ sites.

DOI
10.1103/physrevb.90.064425

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Projected entangled pair states (PEPS) are a promising ansatz for the study of strongly correlated quantum many-body systems in two dimensions. However, due to their high computational cost, developing and improving PEPS algorithms is necessary to make the ansatz widely usable in practice. Here we analyze several algorithmic aspects of the method. On the one hand, we quantify the connection between the correlation length of the PEPS and the accuracy of its approximate contraction and discuss how purifications can be used in the latter. On the other hand, we present algorithmic improvements for the update of the tensor that introduce drastic gains in the numerical conditioning and the efficiency of the algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with the Heisenberg and quantum Ising models on lattices of up to 21\ifmmode×\else\texttimes\fi2121\ifmmode\times\else\texttimes\fi{}2121\ifmmode×\else\texttimes\fi21 sites.
Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension DDD, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to D=10D=10D=10 without resorting to any symmetry. Benchmark tests of the method on the ${J}_{1}\ensuremath{-}{J}_{2}$ model give impressive accuracy compared with exact results.
The approximate contraction of a tensor network of projected entangled pair states (PEPS) is a fundamental ingredient of any PEPS algorithm, required for the optimization of the tensors in ground state search or time evolution, as well as for the evaluation of expectation values. An exact contraction is in general impossible, and the choice of the approximating procedure determines the efficiency and accuracy of the algorithm. We analyze different previous proposals for this approximation, and show that they can be understood via the form of their environment, i.e. the operator that results from contracting part of the network. This provides physical insight into the limitation of various approaches, and allows us to introduce a new strategy, based on the idea of clusters, that unifies previous methods. The resulting contraction algorithm interpolates naturally between the cheapest and most imprecise and the most costly and most precise method. We benchmark the different algorithms with finite PEPS, and show how the cluster strategy can be used for both the tensor optimization and the calculation of expectation values. Additionally, we discuss its applicability to the parallelization of PEPS and to infinite systems.
The authors present a new method for variationally optimizing projected entangled-pair states (PEPS) in the thermodynamic limit based on the global energy functional and novel tensor contraction strategies. It is successfully benchmarked using Ising and Heisenberg Hamiltonians. The method is completely independent from previous PEPS implementations and improves on the variational energies and order parameters as compared to other state-of-the-art PEPS simulations. The algorithm presented here should prove valuable in simulating ground-state properties of strongly correlated quantum spin systems, can be used with other variational ansatzes, and can be applied to tensor network states in general.
The infinite projected entangled pair states (iPEPS) algorithm [J. Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)] has become a useful tool in the calculation of ground-state properties of two-dimensional quantum lattice systems in the thermodynamic limit. Despite its many successful implementations, the method has some limitations in its present formulation which hinder its application to some highly entangled systems. The purpose of this paper is to unravel some of these issues, in turn enhancing the stability and efficiency of iPEPS methods. For this, we first introduce the fast full update scheme, where effective environment and iPEPS tensors are both simultaneously updated (or evolved) throughout time. As we shall show, this implies two crucial advantages: (i) dramatic computational savings and (ii) improved overall stability. In addition, we extend the application of the local gauge fixing, successfully implemented for finite-size PEPS [M. Lubasch et al., Phys. Rev. B 90, 064425 (2014)], to the iPEPS algorithm. We see that the gauge fixing not only further improves the stability of the method but also accelerates the convergence of the alternating least-squares sweeping in the (either ``full'' or ``fast full'') tensor update scheme. The improvement in terms of computational cost and stability of the resulting ``improved'' iPEPS algorithm is benchmarked by studying the ground-state properties of the quantum Heisenberg and transverse-field Ising models on an infinite square lattice.
In the numerical calculation, the projected entangled pair state (PEPS) algorithm is the most important tensor network algorithm for two-dimensional strongly correlated electron quantum lattice system. In this paper, the optimization of PEPS for two-dimensional quantum system is discussed. An optimization connection between how to update the PEPS tensor and how to calculate the physical observable is investigated, for the tensor network algorithm based on the PEPS representation, which can greatly improve the utilization of computing resources. In this case, optimized PEPS algorithm, as a powerful tool, can be used to study quantum phase transitions and quantum critical phenomena in the thermodynamic limit of the two-dimensional strongly correlated electron quantum lattice system. Of course, optimization of PEPS algorithm program has many other applications, such as adding U(1) and SO(2) symmetry in PEPS algorithm, etc.
Numerical treatment of two-dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground-state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this paper, we show how a reformulation of nnn-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model illustrate the improved precision, efficiency, and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.