Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.09326

Probing the Dynamical Structure Factor of Quantum Spin Chains via Low-Temperature Gibbs States with Matrix Product State Subspace Expansion

Tomoya Takahashi, Wei-Lin Tu, Ji-Yao Chen, Yusuke Nomura

Published Jan 14, 2026• 0 citations
arXivHTMLPDF

Abstract

Studying finite-temperature properties with tensor networks is notoriously difficult, especially at low temperatures, due to the rapid growth of entanglement and the complexity of thermal states. Existing methods like purification and minimally entangled typical thermal states offer partial solutions but struggle with scalability and accuracy in low-temperature regime. To overcome these limitations, we propose a new approach based on generating-function matrix product states (GFMPS). By directly computing a large set of Bloch-type excited states, we construct Gibbs states that moderate the area-law constraint, enabling accurate and efficient approximation of low-temperature thermal behavior. Our benchmark results show magnificent agreement with both exact diagonalization and experimental observations, validating the accuracy of our approach. This method offers a promising new direction for overcoming the longstanding challenges of studying low-temperature properties within the tensor network framework. We also expect that our method will facilitate the numerical simulation of quantum materials in comparison with experimental observations.

DOI
10.48550/arxiv.2601.09326

Related papers

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

Chia-Min Chung, Ulrich Schollwöck

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.

arXiv:1910.03329•Oct 8, 2019•2 citations

Finite-Temperature Simulations of Quantum Lattice Models with Stochastic Matrix Product States

Jianxin Gao, Gao Yuan, Qiaoyi Li +1 more

In this work, we develop a stochastic matrix product state (stoMPS) approach that combines the MPS technique and Monte Carlo samplings and can be applied to simulate quantum lattice models down to low temperature. In particular, we exploit a procedure to unbiasedly sample the local tensors in the matrix product states, which has one physical index of dimension $d$ and two geometric indices of dimension $D$, and find the results can be continuously improved by enlarging $D$. We benchmark the methods on small system sizes and then compare the results to those obtained with minimally entangled typical thermal states, finding that stoMPS has overall better performance with finite $D$. We further exploit the MPS sampling to simulate long spin chains, as well as the triangular and square lattices with cylinder circumference $W$ up to 4. Our results showcase the accuracy and effectiveness of stochastic tensor networks in finite-temperature simulations.

arXiv:2312.04420•Dec 7, 2023•0 citations

Gibbs state sampling via cluster expansions

Norhan M. Eassa, Mahmoud M. Moustafa, Arnab Banerjee +1 more

Gibbs states (i.e., thermal states) can be used for several applications such as quantum simulation, quantum machine learning, quantum optimization, and the study of open quantum systems. Moreover, semi-definite programming, combinatorial optimization problems, and training quantum Boltzmann machines can all be addressed by sampling from well-prepared Gibbs states. With that, however, comes the fact that preparing and sampling from Gibbs states on a quantum computer are notoriously difficult tasks. Such tasks can require large overhead in resources and/or calibration even in the simplest of cases, as well as the fact that the implementation might be limited to only a specific set of systems. We propose a method based on sampling from a quasi-distribution consisting of tensor products of mixed states on local clusters, i.e., expanding the full Gibbs state into a sum of products of local “Gibbs-cumulant” type states easier to implement and sample from on quantum hardware. We begin with presenting results for 4-spin linear chains with XY spin interactions, for which we obtain the Z Z dynamical spin-spin correlation functions and dynamical structure factor. We also present the results of measuring the specific heat of the 8-spin chain Gibbs state ρ 8 .

arXiv:2310.20129•npj Quantum Information•Oct 4, 2024•2 citations

Low-temperature Gibbs states with tensor networks

Denise Cocchiarella, Mari Carmen Bañuls

We introduce a tensor network method for approximating thermal equilibrium states of quantum many-body systems at low temperatures. Whereas the usual approach starts from infinite temperature and evolves the state in imaginary time (toward lower temperature), our ansatz is constructed from the zero-temperature limit, the ground state, which can be found with a standard tensor network approach. Motivated by properties of the ground state for conformal field theories, our ansatz is especially well suited near criticality. Moreover, it allows an efficient computation of thermodynamic quantities and entanglement properties. We demonstrate the performance of our approach with a tree tensor network ansatz, although it can be extended to other tensor networks, and present results illustrating its effectiveness in capturing the finite-temperature properties in one- and two-dimensional scenarios. In particular, in the critical one-dimensional case we show how the ansatz reproduces the finite temperature scaling of entanglement in a conformal field theory.

arXiv:2501.08300•Physical Review Research•Jul 29, 2025•5 citations

Generating Function for Projected Entangled-Pair States

Wei-Lin Tu, Laurens Vanderstraeten, Norbert Schuch +3 more

Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin- 1 / 2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin- 1 / 2 J 1 - J 2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations. Published by the American Physical Society 2024

arXiv:2307.08083•PRX Quantum•Mar 1, 2024•16 citations

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
Studying finite-temperature properties with tensor networks is notoriously difficult, especially at low temperatures, due to the rapid growth of entanglement and the complexity of thermal states. Existing methods like purification and minimally entangled typical thermal states offer partial solutions but struggle with scalability and accuracy in low-temperature regime. To overcome these limitations, we propose a new approach based on generating-function matrix product states (GFMPS). By directly computing a large set of Bloch-type excited states, we construct Gibbs states that moderate the area-law constraint, enabling accurate and efficient approximation of low-temperature thermal behavior. Our benchmark results show magnificent agreement with both exact diagonalization and experimental observations, validating the accuracy of our approach. This method offers a promising new direction for overcoming the longstanding challenges of studying low-temperature properties within the tensor network framework. We also expect that our method will facilitate the numerical simulation of quantum materials in comparison with experimental observations.
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.
In this work, we develop a stochastic matrix product state (stoMPS) approach that combines the MPS technique and Monte Carlo samplings and can be applied to simulate quantum lattice models down to low temperature. In particular, we exploit a procedure to unbiasedly sample the local tensors in the matrix product states, which has one physical index of dimension ddd and two geometric indices of dimension DDD, and find the results can be continuously improved by enlarging DDD. We benchmark the methods on small system sizes and then compare the results to those obtained with minimally entangled typical thermal states, finding that stoMPS has overall better performance with finite DDD. We further exploit the MPS sampling to simulate long spin chains, as well as the triangular and square lattices with cylinder circumference WWW up to 4. Our results showcase the accuracy and effectiveness of stochastic tensor networks in finite-temperature simulations.
Gibbs states (i.e., thermal states) can be used for several applications such as quantum simulation, quantum machine learning, quantum optimization, and the study of open quantum systems. Moreover, semi-definite programming, combinatorial optimization problems, and training quantum Boltzmann machines can all be addressed by sampling from well-prepared Gibbs states. With that, however, comes the fact that preparing and sampling from Gibbs states on a quantum computer are notoriously difficult tasks. Such tasks can require large overhead in resources and/or calibration even in the simplest of cases, as well as the fact that the implementation might be limited to only a specific set of systems. We propose a method based on sampling from a quasi-distribution consisting of tensor products of mixed states on local clusters, i.e., expanding the full Gibbs state into a sum of products of local “Gibbs-cumulant” type states easier to implement and sample from on quantum hardware. We begin with presenting results for 4-spin linear chains with XY spin interactions, for which we obtain the Z Z dynamical spin-spin correlation functions and dynamical structure factor. We also present the results of measuring the specific heat of the 8-spin chain Gibbs state ρ 8 .
We introduce a tensor network method for approximating thermal equilibrium states of quantum many-body systems at low temperatures. Whereas the usual approach starts from infinite temperature and evolves the state in imaginary time (toward lower temperature), our ansatz is constructed from the zero-temperature limit, the ground state, which can be found with a standard tensor network approach. Motivated by properties of the ground state for conformal field theories, our ansatz is especially well suited near criticality. Moreover, it allows an efficient computation of thermodynamic quantities and entanglement properties. We demonstrate the performance of our approach with a tree tensor network ansatz, although it can be extended to other tensor networks, and present results illustrating its effectiveness in capturing the finite-temperature properties in one- and two-dimensional scenarios. In particular, in the critical one-dimensional case we show how the ansatz reproduces the finite temperature scaling of entanglement in a conformal field theory.
Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin- 1 / 2 transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin- 1 / 2 J 1 - J 2 model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations. Published by the American Physical Society 2024
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.