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Computer Physics Communications

TeNeS-v2: Enhancement for real-time and finite temperature simulations of quantum many-body systems

Yuichi Motoyama, Tsuyoshi Okubo, Kazuyoshi Yoshimi, Satoshi Morita, Tatsumi Aoyama, Takeo Kato, Naoki Kawashima

Published May 30, 2025• 2 citations
Publisher

Abstract

Quantum many-body systems are challenging targets for computational physics due to their large number of degrees of freedom. The tensor networks, particularly Tensor Product States (TPS) and Projected Entangled Pair States (PEPS), effectively represent these systems on two-dimensional lattices. However, the technical complexity of TPS/PEPS-based coding can be challenging for many researchers to manage effectively. To reduce this problem, we developed TeNeS (Tensor Network Solver). This paper introduces TeNeS -v2, which extends TeNeS with real-time and finite temperature simulations, providing deeper insights into quantum many-body systems. We detail the new algorithms, input/output design, and application examples, demonstrating TeNeS -v2's applicability to various quantum spin and Bose models on two-dimensional lattices. New version program summary Program Title: TeNeS CPC Library link to program files: https://doi.org/10.17632/psm26xxbvd.2 Developer's repository link:: https://github.com/issp-center-dev/TeNeS Licensing provisions: GPLv3 Programming language: C++11 Journal reference of previous version: Comput. Phys. Commun. (2022) 279, 108437, doi:10.1016/j.cpc.2022.108437 Reasons for the new version: To extend the capabilities of TeNeS to include real-time and finite temperature simulations, enabling deeper insights into quantum many-body systems beyond ground states. Summary of revisions: TeNeS -v2 introduces real-time evolution and finite temperature simulation capabilities, expanding the range of quantum many-body phenomena that can be studied. Nature of problem: Quantum many-body systems are extremely difficult to simulate because of the huge dimension of the Hilbert space. Conventional methods have difficulty in accurately representing these systems, especially for properties beyond small lattices and ground states. Advanced computational techniques are required to efficiently capture the dynamics and thermal properties of these systems. Solution method: TeNeS employs tensor networks, specifically Tensor Product States (TPS) and Projected Entangled Pair States (PEPS), to efficiently represent quantum many-body states on two-dimensional lattices. The real-time evolution is handled through a straightforward extension of imaginary time evolution methods, allowing the study of dynamical properties. Finite temperature simulations are conducted using an imaginary time evolution approach starting from the infinite-temperature mixed state. These methods enable the accurate calculation of various physical properties in different quantum states. Additional comments including restrictions and unusual features: TeNeS -v2 maintains ease of use by providing flexible input file configurations and supporting a variety of two-dimensional lattice models. Increasing the bond dimension to improve accuracy requires more computational resources. Finite temperature calculations may exhibit unphysical behavior due to limitations of the tensor network approximation method. Users should be aware of these potential problems and interpret the results following the suggestions made in the text.

DOI
10.1016/j.cpc.2025.109692

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Quantum many-body systems are challenging targets for computational physics due to their large number of degrees of freedom. The tensor networks, particularly Tensor Product States (TPS) and Projected Entangled Pair States (PEPS), effectively represent these systems on two-dimensional lattices. However, the technical complexity of TPS/PEPS-based coding can be challenging for many researchers to manage effectively. To reduce this problem, we developed TeNeS (Tensor Network Solver). This paper introduces TeNeS -v2, which extends TeNeS with real-time and finite temperature simulations, providing deeper insights into quantum many-body systems. We detail the new algorithms, input/output design, and application examples, demonstrating TeNeS -v2's applicability to various quantum spin and Bose models on two-dimensional lattices. New version program summary Program Title: TeNeS CPC Library link to program files: https://doi.org/10.17632/psm26xxbvd.2 Developer's repository link:: https://github.com/issp-center-dev/TeNeS Licensing provisions: GPLv3 Programming language: C++11 Journal reference of previous version: Comput. Phys. Commun. (2022) 279, 108437, doi:10.1016/j.cpc.2022.108437 Reasons for the new version: To extend the capabilities of TeNeS to include real-time and finite temperature simulations, enabling deeper insights into quantum many-body systems beyond ground states. Summary of revisions: TeNeS -v2 introduces real-time evolution and finite temperature simulation capabilities, expanding the range of quantum many-body phenomena that can be studied. Nature of problem: Quantum many-body systems are extremely difficult to simulate because of the huge dimension of the Hilbert space. Conventional methods have difficulty in accurately representing these systems, especially for properties beyond small lattices and ground states. Advanced computational techniques are required to efficiently capture the dynamics and thermal properties of these systems. Solution method: TeNeS employs tensor networks, specifically Tensor Product States (TPS) and Projected Entangled Pair States (PEPS), to efficiently represent quantum many-body states on two-dimensional lattices. The real-time evolution is handled through a straightforward extension of imaginary time evolution methods, allowing the study of dynamical properties. Finite temperature simulations are conducted using an imaginary time evolution approach starting from the infinite-temperature mixed state. These methods enable the accurate calculation of various physical properties in different quantum states. Additional comments including restrictions and unusual features: TeNeS -v2 maintains ease of use by providing flexible input file configurations and supporting a variety of two-dimensional lattice models. Increasing the bond dimension to improve accuracy requires more computational resources. Finite temperature calculations may exhibit unphysical behavior due to limitations of the tensor network approximation method. Users should be aware of these potential problems and interpret the results following the suggestions made in the text.
Quantum many-body systems are challenging targets for computational physics due to their large degrees of freedom. The tensor networks, particularly Tensor Product States (TPS) and Projected Entangled Pair States (PEPS), effectively represent these systems on two-dimensional lattices. However, the technical complexity of TPS/PEPS-based coding is often too much for researchers to handle. To reduce this problem, we developed TeNeS (Tensor Network Solver). This paper introduces TeNeS-v2, which extends TeNeS with real-time and finite temperature simulations, providing deeper insights into quantum many-body systems. We detail the new algorithms, input/output design, and application examples, demonstrating TeNeS-v2's applicability to various quantum spin and Bose models on two-dimensional lattices.
TeNeS (Tensor Network Solver) [1], [2] is a free/libre open-source software program package for calculating two-dimensional many-body quantum states based on the tensor network method and the corner transfer matrix renormalization group (CTMRG) method. This package calculates ground-state wavefunctions for user-defined Hamiltonians and evaluates user-defined physical quantities such as magnetization and correlation functions. For certain predefined models and lattices, there is a tool that makes it easy to generate input files. TeNeS uses an OpenMP/MPI hybrid parallelized tensor operation library and thus can perform large-scale calculations using massively parallel machines. Program Title: TeNeS CPC Library link to program files: https://doi.org/10.17632/psm26xxbvd.1 Code Ocean capsule: https://codeocean.com/capsule/1510058 Licensing provisions: GNU General Public License version 3 Programming language: C++ and python3 External routines/libraries: mptensor Nature of problem: TeNeS calculates the approximate ground states and their properties of user-defined two-dimensional quantum lattice models using user-friendly input files. Numerically exact solutions of such tasks generally require an exponentially diverging computational time, whereas the error in the output of TeNeS is well controlled and can be reduced with a polynomial cost. Solution method: TeNeS implements the tensor networks method based on a tensor-product-state (TPS) wavefunction and the corner transfer matrix renormalization group method. TeNeS also supports massively parallel computing using the library mptensor, which implements OpenMP/MPI hybrid parallelized tensor operations.
The two-dimensional (2D) Hubbard model, widely believed to capture the essential physics of high-T_c cuprate compounds, has attracted immense research interest ever since the discovery of superconductivity in these materials. Despite its superficial simplicity, the Hubbard model incubates abundant fascinating phenomena owing to the strong interactions between charge carriers, encompassing antiferromagnetism, the pseudogap, and unconventional superconductivity. Yet these very interactions induce strong correlations, rendering canonical perturbative approaches unreliable. With advances in modern computational hardware, numerical techniques — notably tensor network methods and Artificial Intelligence (AI) algorithms — have become indispensable to our expedition. Tensor networks encode the amplitude information of many-body quantum states in a network of interconnected tensors. Leveraging the entanglement area law, tensor networks provide efficient, systematic, and controllable representations of quantum states that would otherwise require an exponentially large number of parameters. Past decades have witnessed the development of various tensor network ansatzes, such as Matrix Product States/Operators (MPS/MPO) for one-dimensional systems and Projected Entangled-Pair States (PEPS) for two-dimensional systems, together with diverse algorithms for optimizing these ansatzes for both zero- and finite temperature scenarios. Over the past years, AI technologies have revolutionized the way we analyze scientific data. The transformer architecture, in particular, has demonstrated exceptional capabilities in the domain of natural language processing. Its attention mechanism excels at capturing long-range correlations in sequential data, making it a promising tool for studying strongly correlated many-body systems. The interpretable nature of the attention mechanism further offers valuable insights into the AI's perception of the underlying physics. In this thesis, we assemble an all-inclusive numerical toolchain for analyzing strongly correlated lattice systems. As a demonstration, we investigate the zero- and finite-temperature properties of the 2D Hubbard model on a square lattice via the infinite PEPS (iPEPS) and the eXponential Tensor Renormalization Group (XTRG), respectively. The native two-dimensional iPEPS tensor network faithfully embodies the entanglement area law and helps settle long-standing debates regarding superconducting order in the ground state of the Hubbard model. The XTRG algorithm efficiently constructs thermal density matrices across a broad temperature range via a highly effective exponential-cooling protocol, thereby enabling detailed probes of pairing and pseudogap phenomena. Afterwards, we generate a comprehensive snapshot dataset which furnishes subsequent training of an optimized encoder-only transformer model. Our novel AI architecture features a Markovian interpretation of the attention design and affords improved parallelism. The collective technological stack opens a brand new avenue for exploring the rich physics of strongly correlated many-body systems.
We present PEPSKit...jl, a Julia package for simulating two-dimensional quantum many-body systems with infinite projected entangled-pair states (iPEPS). PEPSKit...jl builds on the TensorKit...jl package for tensor computations and provides high-level algorithms for iPEPS simulations that support both Abelian and non-Abelian symmetries, as well as fermionic systems. This work gives an overview of the main package features, which include support for ground-state, time-evolution, and finite-temperature simulations in systems with different physical symmetries and lattice geometries. These capabilities are illustrated through various examples and technical benchmarks.
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States (PEPS) are well-suited for key classes of physical systems and quantum circuits. However, direct contraction of PEPS networks has exponential cost, while approximate algorithms require computations with large tensors. We propose new scalable algorithms and software abstractions for PEPS-based methods, accelerating the bottleneck operation of contraction and refactorization of a tensor subnetwork. We employ randomized SVD with an implicit matrix to reduce cost and memory footprint asymptotically. Further, we develop a distributed-memory PEPS library and study accuracy and efficiency of alternative algorithms for PEPS contraction and evolution on the Stampede2 supercomputer. We also simulate a popular near-term quantum algorithm, the Variational Quantum Eigensolver (VQE), and benchmark Imaginary Time Evolution (ITE), which compute ground states of Hamiltonians.
Simulating strongly-correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state (PEPS) to represent the vectorization of the quantum thermal state and utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the 2D antiferromagnetic Heisenberg model, the J1J_1J1​-J2J_2J2​ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion (SSE), exact diagonalization, and determinant quantum Monte Carlo (DQMC).