tensor_LO_optim

Code accompanying the paper "Optimal local oscillators for the homodyne detection of multiphoton states"

Overview

This repository provides a tensor-based framework for finding optimal local oscillator (LO) modes for homodyne detection of multiphoton number states. It includes tools for:

• Constructing and decomposing joint spectral amplitude (JSA) tensors for Gaussian and higher-order parametric downconversion (HOPDC) sources using randomized SVD / HOSVD.
• Deriving and numerically testing eigenvalue bounds and conditions relevant to these JSA tensors.
• Comparing the fidelity between the optimal local oscillator and the leading HOSVD mode.
• Optimizing local oscillator modes via a gradient-descent scheme with a hybrid penalty and homotopy continuation method.

Installation

git clone https://github.com/polyquantique/tensor_LO_optim.git
cd tensor_lo_optim
pip install -e .

Dependencies

numpy
scipy
matplotlib
jax
jaxopt
pandas
seaborn
tensorrsvd @ git+https://github.com/PaulVirally/TensorRSVD

These are listed in pyproject.toml and will be installed automatically with pip install -e ..

Repository structure

src/optimization_core/ — main library

The core package implementing the optimization:

• gaussian_jsa.py: construction of Gaussian JSA tensors.
• waveguide_spdc_jsa.py: construction of JSA tensors for higher-order parametric downconversion (HOPDC) in waveguides.
• objectives.py: objective functions used to optimize the local oscillator mode.
• optimizers.py: optimization routines, including the penalty routine, the hybrid penalty and homotopy continuation routine with gradient descent, and trust-region methods.
• run_optimization.py: functions to launch an optimization run end to end using the optimizers previously defined.
• utils.py: helper for loading and saving data, and complex vectors handling.
• bounds.py: functions to calculate the homodyne visibility upper bounds.
• Additional modules with complementary plotting functions for visualizing JSAs, modes, and optimization results.

optimization_experiments/ — examples and paper figures

• 0_example_usage_random_svd/ Example usage of randomized SVD applied to Gaussian JSAs and to JSAs associated with higher-order parametric downconversion (HOPDC).

• 1_mode_decompositions/ Mode decompositions of Gaussian and HOPDC JSAs using randomized SVD.

• 2_Motivating example_initialization/ Example illustrating the motivation behind the initialization strategy used in the optimization routines.

• 3_1_tensor_eigenvalues_gaussians/ Calculation of bounds and tests of the eigenvalue condition for Gaussian JSA tensors.

• 3_1_tensor_eigenvalues_hopdc/ Calculation of bounds and tests of the eigenvalue condition for HOPDC JSA tensors.

• 3_1_tensor_eigenvalues_hopdc_chirped_pump/ Calculation of bounds and tests of the eigenvalue condition for HOPDC JSA tensors with a chirped pump.

• 4_optimal_mode_distributions_fidelity/ Calculation of the fidelity between the optimal local oscillator and the leading HOSVD mode.

• 5_computation/ Examples of the computation of optimal local oscillators for homodyne detection of Gaussian and HOPDC JSAs:

  • Optimization using a hybrid penalty method and homotopy continuation on a gradient-descent method.
  • Examples/comparison with a trust-region method.
  • Example of an optimization/continuation schedule.

  Citation

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

@article{tensor_lo_optim,
  title   = {Optimal local oscillators for the homodyne detection of multiphoton states},
  author  = {Osorio, Gisell Lorena and Virally, Paul and  Molesky, Sean and  Quesada, Nicol{\'a}s},
  year    = {2026},
  doi     = {https://doi.org/10.48550/arXiv.2609.28133},
  url     = {https://arxiv.org/abs/2609.28133},
  publisher = {arXiv}
}

License

This project is licensed under the Apache License 2.0 — see the LICENSE file for details.