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arXiv:1310.8023 · Physical Review B

Improving the efficiency of variational tensor network algorithms

Glen Evenbly, Robert N. C. Pfeifer

Published Jun 12, 2014• 29 citations
arXivHTMLPDF

Abstract

We present several results relating to the contraction of generic tensor networks and discuss their application to the simulation of quantum many-body systems using variational approaches based upon tensor network states. Given a closed tensor network $\mathcal{T}$, we prove that if the environment of a single tensor from the network can be evaluated with computational cost $\ensuremath{\kappa}$, then the environment of any other tensor from $\mathcal{T}$ can be evaluated with identical cost $\ensuremath{\kappa}$. Moreover, we describe how the set of all single tensor environments from $\mathcal{T}$ can be simultaneously evaluated with fixed cost $3\ensuremath{\kappa}$. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a multiscale entanglement renormalization Ansatz for the ground state of a one-dimensional quantum system, where they are shown to substantially reduce the computation time.

DOI
10.1103/physrevb.89.245118

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We present several results relating to the contraction of generic tensor networks and discuss their application to the simulation of quantum many-body systems using variational approaches based upon tensor network states. Given a closed tensor network T\mathcal{T}T, we prove that if the environment of a single tensor from the network can be evaluated with computational cost \ensuremathκ\ensuremath{\kappa}\ensuremathκ, then the environment of any other tensor from T\mathcal{T}T can be evaluated with identical cost \ensuremathκ\ensuremath{\kappa}\ensuremathκ. Moreover, we describe how the set of all single tensor environments from T\mathcal{T}T can be simultaneously evaluated with fixed cost 3\ensuremathκ3\ensuremath{\kappa}3\ensuremathκ. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a multiscale entanglement renormalization Ansatz for the ground state of a one-dimensional quantum system, where they are shown to substantially reduce the computation time.
Variational tensor network optimization has become a powerful tool for studying classical statistical models in two dimensions. However, its application to three-dimensional systems remains limited, primarily due to the high computational cost associated with evaluating the free-energy density and its gradient. This process requires contracting a triple-layer tensor network composed of a projected entangled pair operator and projected entangled pair states. In this paper, we employ a split corner-transfer renormalization group scheme tailored for the contraction of such a triple-layer network, which reduces the computational complexity while keeping high accuracy. Through numerical benchmarks on the three-dimensional classical Ising model, we demonstrate that the proposed scheme achieves numerical results comparable to the most recent Monte Carlo simulations, providing a substantial speedup over previous variational tensor network approaches. This makes this method well-suited for efficient gradient-based optimization in three-dimensional tensor network simulations.
Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g., as in projected entangled pair states, then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining an ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this method to different models and observe favorable energies and runtimes when compared with standard tensor network methods.
In the tensor-network framework, the expectation values of two-dimensional quantum states are evaluated by contracting a double-layer tensor network constructed from initial and final tensor-network states. The computational cost of carrying out this contraction is generally very high, which limits the largest bond dimension of tensor-network states that can be accurately studied to a relatively small value. We propose an optimized contraction scheme to solve this problem by mapping the double-layer tensor network onto an intersected single-layer tensor network. This reduces greatly the bond dimensions of local tensors to be contracted and improves dramatically the efficiency and accuracy of the evaluation of expectation values of tensor-network states. It almost doubles the largest bond dimension of tensor-network states whose physical properties can be efficiently and reliably calculated, and it extends significantly the application scope of tensor-network methods.
Strongly correlated quantum systems are one of the most interesting and challenging phenomena in modern condensed matter physics. Unfortunately, the theoretical access even to their low energy properties is limited by the exponential growth of the Hilbert space in a many-body setting. To mitigate this computational barrier, tensor network methods are one of the most competitive approaches to classically simulate complex quantum systems. In this thesis we will discuss recent developments in the field of tensor networks to efficiently and accurately simulate two-dimensional quantum many-body states in the thermodynamic limit. Hereby, we introduce the variational update globally optimizing the wave-function, encoded as tensor network in the form of an infinite projected entangled pair state (iPEPS), in order to obtain ground states of many-body Hamiltonian models. We focus in the discussion on the well-established method to approximately contract this two-dimensional structure, the corner transfer matrix renormalization group (CTMRG), and discuss its native application on square lattice and triangular lattice based models. Furthermore, we report the development of a refined numerical algorithm, the split-CTMRG, to significantly reduce its computational complexity, pushing the limits of feasible simulations within modern hardware capabilities. Finally, the method advancements are applied to a currently widely studied model of frustrated magnetism, the Heisenberg maple-leaf antiferromagnet. We are able to find a large phase of vanishing magnetization in contradiction to previous works and, to the best of our knowledge, the lowest reported variational ground state energy at the isotropic point of the model. As an outlook, we describe a potential adaption of the previously developed methods to variationally simulate two-dimensional many-body thermal states.
In this thesis, we use the new numerical method multiscale entanglement renormalization ansatz (MERA) to study the one dimensional spin system. This method, MERA, utilizes a network of tensors to represent quantum many-body systems on a lattice. Other examples of tensor network state are matrix product states (MPSs) for 1D systems, tree tensor networks (TTNs) for systems with a tree shape, and projected entangled-pair states (PEPSs) for 2D systems and be- yond. The three structures dier in the graph that denes how the tensors are interconnected into a network: The graphs for MPSs, TTNs, and 2D PEPSs are, respectively, a chain, a tree, and a 2D lattice. Importantly, from these tensor networks the expectation value of local observables can be computed eciently. Developing algorithms to simulate quantum many-body systems is important for understanding and studying the physical principles. And these may become powerful tools for the analysis of quantum computation and quantum information theory. But in general ecient simulation is dicult, due to the fundamental quantum physical laws. The main method we have studied and worked on is multiscale entangle- ment renormalization ansatz (MERA) which is a variational ansatz for many- body states. MERA is a tree tensor network for 1D, 2D systems and beyond. We have developed code of ternary-MERA to compute the ground state energy, expected values of local observables, and correlators for 1D lattice systems. In a nutshell, MERA is an algorithm for entanglement renormalization which is a numerical technique based on locally reorganizing the Hilbert space of a quantum many-body system with the aim to reduce the amount of entanglement in its wave function. We use this method to study 1D Ising model, XXZ model, and J1 - J2 model, and then comparing the numerical results with exact diagonalization (ED) and density matrix renormalization group (DMRG).
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States (PEPS) are well-suited for key classes of physical systems and quantum circuits. However, direct contraction of PEPS networks has exponential cost, while approximate algorithms require computations with large tensors. We propose new scalable algorithms and software abstractions for PEPS-based methods, accelerating the bottleneck operation of contraction and refactorization of a tensor subnetwork. We employ randomized SVD with an implicit matrix to reduce cost and memory footprint asymptotically. Further, we develop a distributed-memory PEPS library and study accuracy and efficiency of alternative algorithms for PEPS contraction and evolution on the Stampede2 supercomputer. We also simulate a popular near-term quantum algorithm, the Variational Quantum Eigensolver (VQE), and benchmark Imaginary Time Evolution (ITE), which compute ground states of Hamiltonians.