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TENSO: Software Package for Numerically Exact Open Quantum Dynamics Based on Efficient Tree Tensor Network Decomposition of the Hierarchical Equations of Motion

Juan C. Rodriguez Betancourt, Michelle C. Anderson, Luchang Niu, Xinxian Chen, Ignacio Franco

Abstract

TENSO is a versatile and powerful open-source software package for numerically exact simulations of the dynamics of quantum systems immersed in structured thermal environments. It is based on a tree tensor network decomposition of the hierarchical equations of motion (HEOM) that efficiently curbs its curse of dimensionality with bath complexity. As such, TENSO enables exact non-Markovian open quantum dynamics simulations even with complex environments typical of chemistry and quantum information science. TENSO allows for time-dependent drive in the system, and for non-commuting fluctuations. More generally, TENSO efficiently propagates the dynamics for any method with a generator of the dynamics that can be expressed in a sum-of-products form, including the HEOM and multi-layer multiconfigurational time-dependent Hartree methods. TENSO enables simulations using tensor trees and trains of arbitrary order, and implements three propagation strategies for the coupled master equations; two fixed-rank methods that require a constant memory footprint during the dynamics and one adaptive rank method with a variable memory footprint controlled by the target level of computational error. In contrast to the accompanying theory and algorithmic paper [J. Chem. Phys. 163, 104109 (2025)] the focus here is on the practical usage and applications of TENSO with underlying theoretical concepts introduced only as needed.

TENSO: Software Package for Numerically Exact Open Quantum Dynamics Based on Efficient Tree Tensor Network Decomposition of the Hierarchical Equations of Motion

Abstract

TENSO is a versatile and powerful open-source software package for numerically exact simulations of the dynamics of quantum systems immersed in structured thermal environments. It is based on a tree tensor network decomposition of the hierarchical equations of motion (HEOM) that efficiently curbs its curse of dimensionality with bath complexity. As such, TENSO enables exact non-Markovian open quantum dynamics simulations even with complex environments typical of chemistry and quantum information science. TENSO allows for time-dependent drive in the system, and for non-commuting fluctuations. More generally, TENSO efficiently propagates the dynamics for any method with a generator of the dynamics that can be expressed in a sum-of-products form, including the HEOM and multi-layer multiconfigurational time-dependent Hartree methods. TENSO enables simulations using tensor trees and trains of arbitrary order, and implements three propagation strategies for the coupled master equations; two fixed-rank methods that require a constant memory footprint during the dynamics and one adaptive rank method with a variable memory footprint controlled by the target level of computational error. In contrast to the accompanying theory and algorithmic paper [J. Chem. Phys. 163, 104109 (2025)] the focus here is on the practical usage and applications of TENSO with underlying theoretical concepts introduced only as needed.
Paper Structure (30 sections, 33 equations, 9 figures, 2 tables)

This paper contains 30 sections, 33 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Bath Correlation Function and its Decomposition into Features
  4. Hierarchical Equations of Motion
  5. Tree Tensor Network Decomposition
  6. Numerical examples
  7. Obtaining and Installing TENSO
  8. Overview
  9. Example 1: Spin-Boson
  10. Dynamics in a structured bath
  11. Comparison of Different Tensor Decompositions
  12. Non-Commuting Fluctuations
  13. Driven Structured Spin-Boson Model
  14. Adaptability of the Framework to Other Equations of Motion
  15. Convergence Parameters
  16. ...and 15 more sections

Figures (9)

  • Figure 1: Structured Spin-Boson Quantum Dynamics. (a) Population relaxation dynamics of the spin-boson system, with the excited state in blue and ground state in orange, showing that dynamics are identical whether the TT or BTT tensor network decomposition is employed. (b) Structured spectral density employed in the simulation.
  • Figure 2: Spin-Boson With Two Baths: Evolution of a spin-boson system coupled to two non-commuting fluctuations.
  • Figure 3: Driven Spin-Boson Dynamics: (a) Time evolution of populations and (b) coherence evolution under a resonant $\pi/2$ Gaussian pulse while interacting with a structured bath. (c) The structured spectral density of the thermal bath. The laser-induced control dynamics demonstrates coherent excitation in the two-level system.
  • Figure 4: MCTDH vs HEOM. Short comparison of an MCTDH and HEOM calculation for a simple Brownian bath at 300 K and pure dephasing.
  • Figure 5: Convergence Demonstration (a) Population of a spin-boson model simulated with the BTT method for several different hierarchy depths demonstrating the spurious oscillations that occur when the depth is insufficient. (b) Population of a spin-boson model simulated with the BTT method for several different ranks showing less sensitivity to rank. (c) Population of a spin-boson model simulated with the TT method for several different ranks showing more sensitivity to rank. (d) Comparison of mean absolute deviation of population from reference population with depth and rank.
  • ...and 4 more figures