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arXiv:1906.12030

TensorNetwork on TensorFlow: Entanglement Renormalization for quantum critical lattice models

Martin Ganahl, Ashley Milsted, Stefan Leichenauer, Jack D. Hidary, Guifré Vidal

Published Jun 28, 2019• 2 citations
arXivHTMLPDF

Abstract

We use TensorNetwork [C. Roberts et al., arXiv: 1905.01330], a recently developed API for performing tensor network contractions using accelerated backends such as TensorFlow, to implement an optimization algorithm for the Multi-scale Entanglement Renormalization Ansatz (MERA). We use the MERA to approximate the ground state wave function of the infinite, one-dimensional transverse field Ising model at criticality, and extract conformal data from the optimized ansatz. Comparing run times of the optimization on CPUs vs. GPU, we report a very significant speed-up, up to a factor of 200, of the optimization algorithm when run on a GPU.

DOI
10.48550/arxiv.1906.12030

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We use TensorNetwork [C. Roberts et al., arXiv: 1905.01330], a recently developed API for performing tensor network contractions using accelerated backends such as TensorFlow, to implement an optimization algorithm for the Multi-scale Entanglement Renormalization Ansatz (MERA). We use the MERA to approximate the ground state wave function of the infinite, one-dimensional transverse field Ising model at criticality, and extract conformal data from the optimized ansatz. Comparing run times of the optimization on CPUs vs. GPU, we report a very significant speed-up, up to a factor of 200, of the optimization algorithm when run on a GPU.
The multi-scale entanglement renormalization ansatz (MERA) provides a natural description of the ground state of a quantum critical Hamiltonian on the lattice. From an optimized MERA, one can extract the scaling dimensions of the underlying conformal field theory. However, extracting conformal data from the ascending superoperator derived from MERA seems to have a limited range of accuracy, even after increasing the bond dimension. Here, we propose an alternative ansatz based on an increasing number of disentangling layers. This leads to generalized versions of MERA that improve the extraction of scaling dimensions, as tested in the Gaussian MERA setting for the critical Ising model.
TensorNetwork is an open source library for implementing tensor network algorithms in TensorFlow. We describe a tree tensor network (TTN) algorithm for approximating the ground state of either a periodic quantum spin chain (1D) or a lattice model on a thin torus (2D), and implement the algorithm using TensorNetwork. We use a standard energy minimization procedure over a TTN ansatz with bond dimension χχχ, with a computational cost that scales as O(χ4)O(χ^4)O(χ4). Using bond dimension χ∈[32,256]χ\in [32,256]χ∈[32,256] we compare the use of CPUs with GPUs and observe significant computational speed-ups, up to a factor of 100100100, using a GPU and the TensorNetwork library.
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).
We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the recently proposed tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] to the Euclidean time evolution operator e(-βH) for infinite β. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature β, produces a MERA representation of a thermal Gibbs state. Our construction endows tensor network renormalization with a renormalization group flow in the space of wave functions and Hamiltonians (and not merely in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.
In this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems. We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization strategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.
Strongly correlated quantum matter, such as the insufficiently understood high- temperature superconductors, are difficult to simulate numerically. The multiscale entanglement renormalization ansatz (MERA) is a hierarchical class of tensor networks that efficiently captures the typical ground-state entanglement structures. Motivated by recent advances to employ MERA in quantum-classical algorithms, the authors take a fresh look here, analyzing new variations of the approach including tensor Trotterization and importance sampling. Optimal contraction sequences and algorithmic phase diagrams show that these quantum-inspired methods can substantially reduce computational costs.