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:2405.18484 · Physical review. B./Physical review. B

Spectroscopy and complex-time correlations using minimally entangled typical thermal states

Zhenjiu Wang, Paul McClarty, Dobromila Dankova, A. Honecker, Alexander Wietek

Published Sep 22, 2025• 2 citations
arXivHTMLPDF

Abstract

Tensor network states have enjoyed great success in capturing aspects of strong correlation physics. However, obtaining dynamical correlators at nonzero temperatures is generically hard even using these methods. Here, we introduce a practical approach to computing such correlators using minimally entangled typical thermal states (METTS). While our primary method directly computes dynamical correlators of physical operators in real time, we propose extensions where correlations are evaluated in the complex-time plane. The imaginary time component bounds the rate of entanglement growth and strongly alleviates the computational difficulty allowing the study of larger system sizes. To extract the physical correlators, one must take the limit of purely real-time evolution. We present two routes for obtaining this information: (i) via an correlation function in complex time combined with a stochastic analytic continuation method to obtain the real-time limit and (ii) a correlation function that asymptotically captures the desired correlation function quantitatively. We show that these numerical techniques capture the finite-temperature dynamics of the Shastry-Sutherland model—a model of interacting spins one-half in two spatial dimensions.

DOI
10.1103/fc58-w1rr

Related papers

Minimally entangled typical thermal state algorithms for finite temperature Matsubara Green functions

Daniel Bauernfeind, Xiaodong Cao, Edwin Miles Stoudenmire +1 more

We extend finite temperature tensor network methods to compute Matsubara imaginary time correlation functions, building on the minimally entangled typical thermal states (METTS) and purification algorithms. While imaginary time correlation functions are straightforward to formulate with these methods, care is needed to avoid convergence issues that would result from naive estimators. As a benchmark, we study the single-band Anderson impurity model, even though the algorithm is quite general and applies to lattice models. The special structure of the impurity model benchmark system and our choice of basis enable techniques such as the reuse of high-probability METTS for increasing algorithm efficiency. The results are competitive with state-of-the-art continuous-time Monte Carlo. We discuss the behavior of computation time and error as a function of the number of purified sites in the Hamiltonian.

arXiv:2107.13941•Physical review. B./Physical review. B•May 6, 2022•3 citations

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

Moritz Binder, Thomas Barthel

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.

arXiv:1411.3033•Physical Review B•Sep 10, 2015•38 citations

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

Quantum time correlation functions from complex time Monte Carlo simulations: A maximum entropy approach

Goran Krilov, Eunji Sim, B. J. Berne

We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.

The Journal of Chemical Physics•Jan 15, 2001•77 citations

Minimally Entangled Typical Thermal States for Classical and Quantum Simulation of 1+1-Dimensional $\mathbb Z_2$ Lattice Gauge Theory at Finite Temperature and Density

Chen, I-Chi, João C. Getelina, Klée Pollock +4 more

Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics that also has fundamental implications for condensed matter physics. In this work, we use minimally entangled typical thermal state (METTS) approaches to facilitate both classical and quantum computational studies of such systems. METTS techniques combine classical random sampling with imaginary time evolution, which can be performed on either a classical or a quantum computer, to estimate thermal averages of observables. We study 1+1-dimensional $\mathbb{Z}_2$ gauge theory coupled to spinless fermionic matter, which maps onto a local quantum spin chain. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach that is a promising candidate for implementation on near-term quantum devices, focusing on the equation of state as well as on various measures of fermion confinement. Of particular importance is the choice of basis for obtaining new METTS samples, which impacts both the classical sampling complexity (a key factor in both classical and quantum simulation applications) and complexity of circuits used in the quantum computing approach. Our work sets the stage for future studies of strongly coupled gauge theories with both classical and quantum hardware.

arXiv:2407.11949•Jul 16, 2024•0 citations

Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations

João C. Getelina, Niladri Gomes, Thomas Iadecola +2 more

Scalable quantum algorithms for the simulation of quantum many-body systems in thermal equilibrium are important for predicting properties of quantum matter at finite temperatures. Here we describe and benchmark a quantum computing version of the minimally entangled typical thermal states (METTS) algorithm for which we adopt an adaptive variational approach to perform the required quantum imaginary time evolution. The algorithm, which we name AVQMETTS, dynamically generates compact and problem-specific quantum circuits, which are suitable for noisy intermediate-scale quantum (NISQ) hardware. We benchmark AVQMETTS on statevector simulators and perform thermal energy calculations of integrable and nonintegrable quantum spin models in one and two dimensions and demonstrate an approximately linear system-size scaling of the circuit complexity. We further map out the finite-temperature phase transition line of the two-dimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model.

arXiv:2301.02592•SciPost Physics•Sep 19, 2023•21 citations
Tensor network states have enjoyed great success in capturing aspects of strong correlation physics. However, obtaining dynamical correlators at nonzero temperatures is generically hard even using these methods. Here, we introduce a practical approach to computing such correlators using minimally entangled typical thermal states (METTS). While our primary method directly computes dynamical correlators of physical operators in real time, we propose extensions where correlations are evaluated in the complex-time plane. The imaginary time component bounds the rate of entanglement growth and strongly alleviates the computational difficulty allowing the study of larger system sizes. To extract the physical correlators, one must take the limit of purely real-time evolution. We present two routes for obtaining this information: (i) via an correlation function in complex time combined with a stochastic analytic continuation method to obtain the real-time limit and (ii) a correlation function that asymptotically captures the desired correlation function quantitatively. We show that these numerical techniques capture the finite-temperature dynamics of the Shastry-Sutherland model—a model of interacting spins one-half in two spatial dimensions.
We extend finite temperature tensor network methods to compute Matsubara imaginary time correlation functions, building on the minimally entangled typical thermal states (METTS) and purification algorithms. While imaginary time correlation functions are straightforward to formulate with these methods, care is needed to avoid convergence issues that would result from naive estimators. As a benchmark, we study the single-band Anderson impurity model, even though the algorithm is quite general and applies to lattice models. The special structure of the impurity model benchmark system and our choice of basis enable techniques such as the reuse of high-probability METTS for increasing algorithm efficiency. The results are competitive with state-of-the-art continuous-time Monte Carlo. We discuss the behavior of computation time and error as a function of the number of purified sites in the Hamiltonian.
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 present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.
Minimally Entangled Typical Thermal States for Classical and Quantum Simulation of 1+1-Dimensional Z2\mathbb Z_2Z2​ Lattice Gauge Theory at Finite Temperature and Density
Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics that also has fundamental implications for condensed matter physics. In this work, we use minimally entangled typical thermal state (METTS) approaches to facilitate both classical and quantum computational studies of such systems. METTS techniques combine classical random sampling with imaginary time evolution, which can be performed on either a classical or a quantum computer, to estimate thermal averages of observables. We study 1+1-dimensional Z2\mathbb{Z}_2Z2​ gauge theory coupled to spinless fermionic matter, which maps onto a local quantum spin chain. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach that is a promising candidate for implementation on near-term quantum devices, focusing on the equation of state as well as on various measures of fermion confinement. Of particular importance is the choice of basis for obtaining new METTS samples, which impacts both the classical sampling complexity (a key factor in both classical and quantum simulation applications) and complexity of circuits used in the quantum computing approach. Our work sets the stage for future studies of strongly coupled gauge theories with both classical and quantum hardware.
Scalable quantum algorithms for the simulation of quantum many-body systems in thermal equilibrium are important for predicting properties of quantum matter at finite temperatures. Here we describe and benchmark a quantum computing version of the minimally entangled typical thermal states (METTS) algorithm for which we adopt an adaptive variational approach to perform the required quantum imaginary time evolution. The algorithm, which we name AVQMETTS, dynamically generates compact and problem-specific quantum circuits, which are suitable for noisy intermediate-scale quantum (NISQ) hardware. We benchmark AVQMETTS on statevector simulators and perform thermal energy calculations of integrable and nonintegrable quantum spin models in one and two dimensions and demonstrate an approximately linear system-size scaling of the circuit complexity. We further map out the finite-temperature phase transition line of the two-dimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model.