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arXiv:2202.04089 · PRX Quantum

Entanglement Estimation in Tensor Network States via Sampling

Noa Feldman, Augustine Kshetrimayum, Jens Eisert, Moshe Goldstein

Published Jul 25, 2022• 17 citations
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Abstract

We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions.Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory.In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames.Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components.We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry.Although the cost is exponential in the subsystem size, it is sufficiently moderate so that-in contrast with other approaches-accurate results can be obtained on a personal computer for relatively large subsystem sizes.

DOI
10.1103/prxquantum.3.030312

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We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions.Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory.In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames.Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components.We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry.Although the cost is exponential in the subsystem size, it is sufficiently moderate so that-in contrast with other approaches-accurate results can be obtained on a personal computer for relatively large subsystem sizes.
Quantifying experimentally created entanglement could in principle be accomplished by measuring the entire density matrix and calculating an entanglement measure of choice thereafter. Due to the tensor structure of the Hilbert space, this approach becomes unfeasible even for medium-sized systems. Here we present methods for quantifying the entanglement of arbitrarily large two-colorable graph states from simple measurements. The presented methods provide non-trivial bounds on the entanglement for any state as long as there is sufficient fidelity with such a graph state. The measurement data considered here is merely given by stabilizer measurements, thus leading to an exponential reduction in the number of measurements required. We provide analytical results for the robustness of entanglement and the relative entropy of entanglement.
We report on tensor renormalization group calculations of entanglement entropy in one-dimensional quantum systems. The reduced density matrix of a Gibbs state can be represented as a 1+11 + 11+1-dimensional tensor network, which is analogous to the tensor network representation of the partition function. The HOTRG method is used to approximate the reduced density matrix for arbitrary subsystem sizes, from which we obtain the entanglement entropy. We test our method in the quantum Ising model and obtain the entanglement entropy of the ground state by taking the size of time direction to infinity. The central charge ccc is obtained as c=0.49997(8)c = 0.49997(8)c=0.49997(8) for a bond dimension Dcut=96D_\text{cut}=96Dcut​=96, which agrees with the theoretical value c=1/2c=1/2c=1/2 within the error.
We develop a random sampling method for calculating the time evolution of the R\'enyi entanglement entropy after a quantum quench from an insulating state in free boson systems. Because of the non-Gaussian nature of the initial state, calculating the R\'enyi entanglement entropy calls for the exponential cost of computing a matrix permanent. We numerically demonstrate that a simple random sampling method reduces the computational cost of a permanent; for an ${N}_{\mathrm{s}}\ifmmode\times\else\texttimes\fi{}{N}_{\mathrm{s}}$ matrix corresponding to Ns{N}_{\mathrm{s}}Ns​ sites at half filling, the sampling cost becomes $O({2}^{\ensuremath{\alpha}{N}_{\mathrm{s}}})$ with a constant $\ensuremath{\alpha}\ensuremath{\ll}1$, in contrast to the conventional algorithm with the O(2Ns)O({2}^{{N}_{\mathrm{s}}})O(2Ns​) number of summations requiring the exponential time cost. Although the computational cost is still exponential, this improvement allows us to obtain the entanglement entropy dynamics in free boson systems for more than 100 sites. We present several examples of the entanglement entropy dynamics in low-dimensional free boson systems.
A bstract We propose a method to compute the entanglement entropy (EE) using the tensor renormalization group (TRG) method. The reduced density matrix of a d -dimensional quantum system is represented as a ( d + 1)-dimensional tensor network. We develop an explicit algorithm for d = 1 that enables the calculation of EE for single-interval subsystems of arbitrary size. We test our method in two-dimensional tensor network of the Ising model. The central charge is obtained as c = 0 . 49997(8) for D = 96, which agrees with the theoretical prediction within an error, demonstrating the accuracy and reliability of our proposed method.
Sampling a quantum system's underlying probability distributions is an important computational task, e.g., for quantum advantage experiments and quantum Monte Carlo algorithms. Tensor networks are an invaluable tool for efficiently representing states of large quantum systems with limited entanglement. Algorithms for sampling one-dimensional (1D) tensor networks are well-established and utilized in several 1D tensor network methods. In this paper we introduce two novel sampling algorithms for two-dimensional (2D) isometric tensor network states (isoTNS) that generalize existing 1D tensor networks sampling algorithms. Our first proposed algorithm performs independent sampling and yields a single configuration together with its associated probability. The second algorithm employs a greedy search strategy to identify KKK high-probability configurations and their corresponding probabilities. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size.
Symmetry and entanglement are two fundamental concepts in quantum many-body physics. Their interplay is captured by symmetry-resolved entanglement, which decomposes the total entanglement into contributions from different symmetry sectors. Computing symmetry-resolved entanglement in strongly interacting higher-dimensional quantum systems remains challenging. Here, we formulate and implement an estimator-based quantum Monte Carlo (QMC) framework for computing symmetry-resolved Rényi entropies (SRRE) in sign-problem-free interacting lattice systems by measuring disorder (symmetry-twisted) operators in ordinary and replica ensembles and reconstructing SRRE from the corresponding charged moments. We validate the framework in two controlled one-dimensional settings: the transverse-field Ising model (TFIM), for which exact conformal-field-theory predictions are available, and the interacting Heisenberg chain, which tests the U(1) U ( 1 ) symmetry-sector reconstruction and its finite-size behavior. We then apply the method to the two-dimensional TFIM. Within the accessible system sizes and a phenomenological finite-size extrapolation, our data provide numerical evidence consistent with entanglement equipartition at the (2+1) ( 2 + 1 ) D Ising critical point. Our work establishes a practical numerical route to symmetry-resolved entanglement in interacting lattice models and provides a framework for future studies beyond one dimension.