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Minimally entangled typical thermal state algorithms

Stoudenmire, E. M., Steven R. White

Published May 28, 2010• 91 citations
OpenAlex

Abstract

We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to ground state DMRG. Detailed implementations of each step of the method are presented, along with efficient algorithms for working with matrix product states and matrix product operators. We furthermore explore how properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.

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Minimally entangled typical thermal state algorithms

E M Stoudenmire, Steven R White

We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to the ground state density matrix renormalization group (DMRG). Detailed implementation of each step of the method is presented, along with efficient algorithms for working with matrix product states and matrix product operators. Furthermore, we explore how the properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.

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Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles

Moritz Binder, Thomas Barthel

Based on the density matrix renormalization group (DMRG), strongly correlated quantum many-body systems at finite temperatures can be simulated by sampling over a certain class of pure matrix product states (MPS) called minimally entangled typical thermal states (METTS). When a system features symmetries, these can be utilized to substantially reduce MPS computation costs. It is conceptually straightforward to simulate canonical ensembles using symmetric METTS. In practice, it is important to alternate between different symmetric collapse bases to decrease autocorrelations in the Markov chain of METTS. To this purpose, we introduce symmetric Fourier and Haar-random block bases that are efficiently mixing. We also show how grand-canonical ensembles can be simulated efficiently with symmetric METTS. We demonstrate these approaches for spin-$1/2$ $XXZ$ chains and discuss how the choice of the collapse bases influences autocorrelations as well as the distribution of measurement values and, hence, convergence speeds.

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Minimally entangled typical thermal states versus matrix product purifications for the simulation of equilibrium states and time evolution

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For the simulation of equilibrium states and finite-temperature response functions of strongly correlated quantum many-body systems, we compare the efficiencies of two different approaches in the framework of the density matrix renormalization group (DMRG). The first is based on matrix product purifications. The second, more recent one, is based on so-called minimally entangled typical thermal states (METTS). For the latter, we highlight the interplay of statistical and DMRG truncation errors, discuss the use of self-averaging effects, and describe schemes for the computation of response functions. For critical as well as gapped phases of the $\text{spin}\ensuremath{-}1/2$ XXZ chain and the one-dimensional Bose-Hubbard model, we assess the computation costs and accuracies of the two methods at different temperatures. For almost all considered cases, we find that, for the same computation cost, purifications yield more accurate results than METTS---often by orders of magnitude. The METTS algorithm becomes more efficient only for temperatures well below the system's energy gap. The exponential growth of the computation cost in the evaluation of response functions limits the attainable time scales in both methods and we find that in this regard, METTS do not outperform purifications.

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Minimally entangled typical thermal states with auxiliary matrix-product-state bases

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Finite temperature problems in the strong correlated systems are important but challenging tasks. Minimally entangled typical thermal states (METTS) are a powerful method in the framework of tensor network methods to simulate finite temperature systems, including Fermions and frustrated spins which introduce a sign problem in the typical Monte Carlo methods. In this work, we introduce an extension of the METTS algorithm by using a new basis, the auxiliary matrix product state. This new basis achieves the pre-summation process in the partition function, and thus improve the convergence in the Monte Carlo samplings. The method also has the advantage of simulating the grand canonical ensemble in a computationally efficient way by employing good quantum numbers. We benchmark our method on the spin-$1/2$ XXZ model on the triangular lattice, and show that the new method outperforms the original METTS as well as the purification methods at sufficiently low temperature, the usual range of applications of METTS. The new method also naturally connects the METTS method to the purification method.

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Efficient simulation of low-temperature physics in one-dimensional gapless systems

Yuya Kusuki, Kotaro Tamaoka, Zixia Wei +1 more

We discuss the computational efficiency of the finite-temperature simulation with minimally entangled typical thermal states (METTS). To argue that METTS can be efficiently represented as matrix product states, we present an analytic upper bound for the average entanglement R\'enyi entropy of METTS for a R\'enyi index $0<q\ensuremath{\le}1$. In particular, for one-dimensional (1D) gapless systems described by conformal field theories, the upper bound scales as $O(c{N}^{0}log\ensuremath{\beta})$ where $c$ is the central charge and $N$ is the system size. Furthermore, we numerically find that the average R\'enyi entropy exhibits a universal behavior characterized by the central charge and is roughly given by half of the analytic upper bound. Based on these results, we show that METTS can provide a speedup compared to employing the purification method to analyze thermal equilibrium states at low temperatures in 1D gapless systems.

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Efficient representation of minimally entangled typical thermal states in two dimensions via projected entangled pair states

Aritra Sinha, Marek M. Rams, Jacek Dziarmaga

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.

arXiv:2310.08533•Physical review. B./Physical review. B•Jan 19, 2024•10 citations
We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to ground state DMRG. Detailed implementations of each step of the method are presented, along with efficient algorithms for working with matrix product states and matrix product operators. We furthermore explore how properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.
We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to the ground state density matrix renormalization group (DMRG). Detailed implementation of each step of the method is presented, along with efficient algorithms for working with matrix product states and matrix product operators. Furthermore, we explore how the properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.
Based on the density matrix renormalization group (DMRG), strongly correlated quantum many-body systems at finite temperatures can be simulated by sampling over a certain class of pure matrix product states (MPS) called minimally entangled typical thermal states (METTS). When a system features symmetries, these can be utilized to substantially reduce MPS computation costs. It is conceptually straightforward to simulate canonical ensembles using symmetric METTS. In practice, it is important to alternate between different symmetric collapse bases to decrease autocorrelations in the Markov chain of METTS. To this purpose, we introduce symmetric Fourier and Haar-random block bases that are efficiently mixing. We also show how grand-canonical ensembles can be simulated efficiently with symmetric METTS. We demonstrate these approaches for spin-1/21/21/2 XXZXXZXXZ chains and discuss how the choice of the collapse bases influences autocorrelations as well as the distribution of measurement values and, hence, convergence speeds.
For the simulation of equilibrium states and finite-temperature response functions of strongly correlated quantum many-body systems, we compare the efficiencies of two different approaches in the framework of the density matrix renormalization group (DMRG). The first is based on matrix product purifications. The second, more recent one, is based on so-called minimally entangled typical thermal states (METTS). For the latter, we highlight the interplay of statistical and DMRG truncation errors, discuss the use of self-averaging effects, and describe schemes for the computation of response functions. For critical as well as gapped phases of the $\text{spin}\ensuremath{-}1/2$ XXZ chain and the one-dimensional Bose-Hubbard model, we assess the computation costs and accuracies of the two methods at different temperatures. For almost all considered cases, we find that, for the same computation cost, purifications yield more accurate results than METTS---often by orders of magnitude. The METTS algorithm becomes more efficient only for temperatures well below the system's energy gap. The exponential growth of the computation cost in the evaluation of response functions limits the attainable time scales in both methods and we find that in this regard, METTS do not outperform purifications.
Finite temperature problems in the strong correlated systems are important but challenging tasks. Minimally entangled typical thermal states (METTS) are a powerful method in the framework of tensor network methods to simulate finite temperature systems, including Fermions and frustrated spins which introduce a sign problem in the typical Monte Carlo methods. In this work, we introduce an extension of the METTS algorithm by using a new basis, the auxiliary matrix product state. This new basis achieves the pre-summation process in the partition function, and thus improve the convergence in the Monte Carlo samplings. The method also has the advantage of simulating the grand canonical ensemble in a computationally efficient way by employing good quantum numbers. We benchmark our method on the spin-1/21/21/2 XXZ model on the triangular lattice, and show that the new method outperforms the original METTS as well as the purification methods at sufficiently low temperature, the usual range of applications of METTS. The new method also naturally connects the METTS method to the purification method.
We discuss the computational efficiency of the finite-temperature simulation with minimally entangled typical thermal states (METTS). To argue that METTS can be efficiently represented as matrix product states, we present an analytic upper bound for the average entanglement R\'enyi entropy of METTS for a R\'enyi index $0<q\ensuremath{\le}1$. In particular, for one-dimensional (1D) gapless systems described by conformal field theories, the upper bound scales as $O(c{N}^{0}log\ensuremath{\beta})$ where ccc is the central charge and NNN is the system size. Furthermore, we numerically find that the average R\'enyi entropy exhibits a universal behavior characterized by the central charge and is roughly given by half of the analytic upper bound. Based on these results, we show that METTS can provide a speedup compared to employing the purification method to analyze thermal equilibrium states at low temperatures in 1D gapless systems.
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.