README: Tensor Network HHL Simulation

Project Title

Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective

This repository accompanies the work presented in the paper:

"Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective" Alejandro Mata Ali and Iñigo Perez Delgado and Marina Ristol Roura and Aitor Moreno Fdez. de Leceta and Sebastián V. Romero (2025) arXiv:2309.05290

It can be consulted online in the Streamlit webpage: https://tensornetworks-hhl-algorithm.streamlit.app/

The project implements a classical simulation of the quantum Harrow-Hassidim-Lloyd (HHL) algorithm using tensor networks and qudit formalism. The goal is to provide a quantum-inspired solver that models the ideal behavior of HHL efficiently on classical hardware, enabling benchmarking and theoretical lower-bound estimations.

---

Files

• tensor_network_HHL.ipynb: Jupyter notebook containing all code, explanations, experiments, and plots. It reproduces the results presented in the paper.

---

Requirements

Install the dependencies with:

pip install numpy matplotlib scipy torch qiskit qiskit_ibm_runtime qiskit_aer

The notebook is compatible with standard Python 3.x and requires no GPU. All computations were tested on CPU.

---

Usage

Open the notebook in Jupyter:

jupyter notebook tensor_network_HHL.ipynb

You may execute all cells sequentially to:

1. Define the tensor operations for the TN-HHL algorithm.

2. Construct tensors for QPE, inversion, and evolution operators.

3. Apply the method to benchmark problems:

   • Forced harmonic oscillator
   • Forced damped oscillator
   • 2D static heat equation with sources

4. Compare TN-HHL performance to:

   • Exact inversion (PyTorch)
   • Qiskit HHL simulation (for small cases)

Each section is self-contained and annotated for clarity.

---

Summary of the Algorithm

• The notebook encodes the HHL quantum circuit using tensor networks.
• It implements all gates (QPE, inversion, unitaries) as tensor contractions.
• Eigenvalue resolution (parameter mu) and time evolution (parameter tau) are tunable.
• The final solution vector $\vec{x}$ is obtained deterministically, bypassing quantum limitations like post-selection.

---

Reproducibility

The notebook is reproducible and self-contained. It includes exact matrix definitions, right-hand sides, and benchmark hyperparameters.

---

Reference

If you use this code, please cite the original paper:

@misc{ali2024solvingsystemslinearequations,
      title={Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective}, 
      author={Alejandro Mata Ali and Iñigo Perez Delgado and Marina Ristol Roura and Aitor Moreno Fdez. de Leceta and Sebastián V. Romero},
      year={2024},
      eprint={2309.05290},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2309.05290}, 
}

---

License

MIT License (c) Alejandro Mata Ali, 2025