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arXiv:2106.00710 · PRX Quantum

Locally Accurate Tensor Networks for Thermal States and Time Evolution

Alvaro Martin Alhambra, J. I. Cirac

Published Nov 10, 2021• 29 citations
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Abstract

Tensor-network methods are routinely used in approximating various equilibrium and nonequilibrium scenarios, with the algorithms requiring a small bond dimension at low enough time or inverse temperature.These approaches so far lacked a rigorous mathematical justification, since existing approximations to thermal states and time evolution demand a bond dimension growing with system size.To address this problem, we construct projected entangled-pair operators that approximate, for all local observables, (i) their thermal expectation values and (ii) their Heisenberg time evolution.The bond dimension required does not depend on system size, but only on the temperature or time.We also show how these can be used to approximate thermal correlation functions and expectation values in quantum quenches.

DOI
10.1103/prxquantum.2.040331

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Tensor-network methods are routinely used in approximating various equilibrium and nonequilibrium scenarios, with the algorithms requiring a small bond dimension at low enough time or inverse temperature.These approaches so far lacked a rigorous mathematical justification, since existing approximations to thermal states and time evolution demand a bond dimension growing with system size.To address this problem, we construct projected entangled-pair operators that approximate, for all local observables, (i) their thermal expectation values and (ii) their Heisenberg time evolution.The bond dimension required does not depend on system size, but only on the temperature or time.We also show how these can be used to approximate thermal correlation functions and expectation values in quantum quenches.
We propose an efficient algorithm for simulating quantum many-body systems in two spatial dimensions using projected entangled pair states. This is done by approximating the environment, arising in the context of updating tensors in the process of time evolution, using a single-layered tensor network structure. This significantly reduces the computational costs and allows simulations in a larger submanifold of the Hilbert space as bounded by the bond dimension of the tensor network. We present numerical evidence for stability of the method on an antiferromagnetic isotropic Heisenberg model where good agreement is found with the available accurate results.
Two-dimensional tensor networks such as projected entangled pairs states (PEPS) are generally hard to contract. This is arguably the main reason why variational tensor network methods in two dimensions are still not as successful as in one dimension. However, this is not necessarily the case if the tensor network represents a gapped ground state of a local Hamiltonian; such states are subject to many constraints and contain much more structure. In this paper, we introduce an approach for approximating the expectation value of a local observable in ground states of local Hamiltonians that are represented by PEPS tensor networks. Instead of contracting the full tensor network, we try to estimate the expectation value using only a local patch of the tensor network around the observable. Surprisingly, we demonstrate that this is often easier to do when the system is frustrated. In such case, the spanning vectors of the local patch are subject to nontrivial constraints that can be utilized via a semidefinite program to calculate rigorous lower and upper bounds on the expectation value. We test our approach in one-dimensional systems, where we show how the expectation value can be calculated up to at least 3 or 4 digits of precision, even when the patch radius is smaller than the correlation length.
We present a general graph-based projected entangled-pair state (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural matrix, which codifies the details of tensor networks on any graphs in any dimension ddd, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., D=100D=100D=100) and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the three-state quantum Potts model in field on the kagome lattice and the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the 2d2d2d quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies.
Tensor network states are for good reasons believed to capture ground states of gapped local Hamiltonians arising in the condensed matter context, states which are in turn expected to satisfy an entanglement area law. However, the computational hardness of contracting projected entangled pair states in two- and higher-dimensional systems is often seen as a significant obstacle when devising higher-dimensional variants of the density-matrix renormalization group method. In this work, we show that for those projected entangled pair states that are expected to provide good approximations of such ground states of local Hamiltonians, one can compute local expectation values in quasipolynomial time. We therefore provide a complexity-theoretic justification of why state-of-the-art numerical tools work so well in practice. We finally turn to the computation of local expectation values on quantum computers, providing a meaningful application for a small-scale quantum computer.
The minimally entangled typical thermal states (METTS) are an ensemble of pure states, equivalent to the Gibbs thermal state, designed with an efficient tensor network representation in mind. In this article, we use the projected entangled pair states (PEPS) as their representation on a two-dimensional (2D) lattice. Unlike matrix product states (MPS), which for 2D systems are limited by an exponential computational barrier in the lattice size, PEPS provides a more tractable approach. To substantiate the prowess of PEPS in modeling METTS (dubbed PEPS-METTS), we benchmark it against the purification method in the context of the 2D quantum Ising model at its critical temperature. Our analysis reveals that PEPS-METTS achieves accurate results with significantly lower bond dimensions. We further corroborate this finding in the 2D Fermi-Hubbard model. At a technical level, we introduce an efficient zipper method to obtain PEPS boundary MPS needed to compute expectation values and perform sampling. The imaginary time evolution is done with the neighborhood tensor update.
We introduce a variational Monte Carlo algorithm for approximating finite-temperature quantum many-body systems, based on the minimization of a modified free energy. This approach directly approximates the state at a fixed temperature, allowing for systematic improvement of the expressiveness without accumulating errors from iterative imaginary-time evolution. We employ a variety of trial states—both tensor networks as well as neural networks—as variational for our numerical optimization. We benchmark and compare different constructions in the above classes, both for one- and two-dimensional problems, with systems made of up to N = 100 spins. Our results demonstrate that while restricted Boltzmann machines show limitations, string bond tensor network states exhibit systematic improvements with increasing bond dimensions and the number of strings.