⚛️ TENSO: Tensor Equations for Non-Markovian Structured Open Systems

📖 Overview

TENSO is a tensor network-based method and software package for generating and integrating master equations for open quantum dynamics in structured thermal environments.

The code is written in Python and leverages PyTorch for efficient, hardware-agnostic tensor operations (running seamlessly on CPUs and GPUs). It is designed to be highly scalable, allowing for the simulation of large open quantum systems by utilizing polynomial-scaling Tree Tensor Network (TTN) topologies instead of exponentially scaling approaches.

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🚀 Quick Setup

Development Installation

1. Create a virtual environment (Python >= 3.10 is required).
2. Ensure core dependencies are available: numpy, scipy, pytorch, torchdiffeq, tqdm.
3. Install TENSO in development mode using pip:

   python -m pip install -e .

4. (Optional) For tutorials and testing, we recommend installing jupyter-lab and matplotlib.

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📚 Documentation & Tutorials

For a detailed guide, API reference, and interactive examples, please visit our official documentation website:

🌐 TENSO Official Documentation

Input files accompanying the TENSO software tutorial can be found in the directory tutorial_scripts/ within this repository.

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🖋️ How to Cite

If you use TENSO in your research, please consider citing our work:

1. The TENSO Software Package

Describes the software implementation and usage tutorials.

Rodriguez-Betancourt, J. C., Anderson, M. C., Niu, L., Chen, X., & Franco, I. (2026). TENSO: Software Package for Numerically Exact Open Quantum Dynamics Based on Efficient Tree Tensor Network Decomposition of the Hierarchical Equations of Motion. Journal of Chemical Theory and Computation, 22(14), 7048-7069. DOI: 10.1021/acs.jctc.6c00525

Show BibTeX

@article{rodriguez2026tenso,
  author  = {Rodriguez-Betancourt, Juan C. and Anderson, Michelle C. and Niu, Luchang and Chen, Xinxian and Franco, Ignacio},
  title   = {{TENSO}: Software Package for Numerically Exact Open Quantum Dynamics Based on Efficient Tree Tensor Network Decomposition of the Hierarchical Equations of Motion},
  journal = {J. Chem. Theory Comput.},
  volume  = {22},
  number  = {14},
  pages   = {7048-7069},
  year    = {2026},
  doi     = {10.1021/acs.jctc.6c00525}
}

2. TTN-HEOM theory and algorithms

Introduces the underlying TTN-HEOM method and time-dependent variational principle.

Chen, X. & Franco, I. (2025). Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments. The Journal of Chemical Physics 163, 104109. DOI: 10.1063/5.0278591

Show BibTeX

@article{Chen2025,
   author = {Xinxian Chen and Ignacio Franco},
   title = {Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments},
   journal = {The Journal of Chemical Physics},
   volume = {163},
   issue = {10},
   pages = {104109},
   year = {2025},
   doi = {10.1063/5.0278591}
}

3. Theoretical Foundation (Bexcitonics)

Develops the bexcitonic quasiparticle picture that underlies the HEOM generalization.

Chen, X. & Franco, I. (2024). Bexcitonics: Quasiparticle approach to open quantum dynamics. The Journal of Chemical Physics, 160(20), 204116. DOI: 10.1063/5.0198567

Show BibTeX

@article{chen2024bexcitonics,
    author = {Chen, Xinxian and Franco, Ignacio},
    title = {Bexcitonics: Quasiparticle approach to open quantum dynamics},
    journal = {The Journal of Chemical Physics},
    volume = {160},
    number = {20},
    pages = {204116},
    year = {2024},
    doi = {10.1063/5.0198567}
}