Generative flow-based warm start of the variational quantum eigensolver

This repository contains the implementation of Flow-VQE, a novel approach combining variational quantum eigensolvers (VQE) with normalizing flows for molecular ground state optimization and warm-start. The code supports the research presented in arXiv:2507.01726.

Overview

The project includes two main components:

• Flow-VQE: Main procedure using normalizing flows as surrogate models for VQE training
• Optimization Baselines: Traditional VQE optimization methods with various optimizers, and post-training by invoking warm-start or parameter-transfer parameters

Installation

1. Install dependencies:

pip install -r requirements.txt

2. Data Repository: Due to data capacity limitations, the experimental results and trained models are stored on the Zenodo Flow-VQE-data-model. After downloading, you can use the trained model to perform the warm-start function.

Quick Start

Flow-VQE Training

Train Flow-VQE models for molecular systems:

# Single distance training for H4
python flow_vqe_main.py --molecules H4 --training_range 2.6 --training_mode single --n_flows 7  --n_epochs 3001

# Multi-distance training for H2O 
python flow_vqe_main.py --molecules H2O --training_range "np.linspace(0.8, 1.8, 6)" --training_mode multi --n_epochs 5001 

Warm-up Parameter Generation

Generate optimized parameters using trained Flow-VQE models:

# Generate warm-up parameters for H2O using the trained flow model 
python flow_vqe_warm_up.py --molecules H2O --test_range "np.linspace(0.75, 1.9, 50)"

Optimization Baselines

Run traditional VQE optimization for different molecules:

# H4 molecule with multiple optimizers  (default bond-length range)
python optimization_baselines_main.py --molecule H4 --mode default --optimizers ADAM GD QNSPSA

# H2O molecule optimization  (default bond-length range)
python optimization_baselines_main.py --molecule H2O --mode default --optimizers ADAM GD QNSPSA

Complete Usage Examples

For detailed usage examples and reproduction of all paper results, please refer to example_usage.py. This file contains:

• All default optimization experiments (Fig. 2)
• Warm-up optimization examples (Table I)
• Parameter transfer comparisons (Fig. 4)
• Flow-VQE training configurations (Fig. 2, 3, 4)
• Complete command-line examples for every experiment in the paper

Simply run:

python example_usage.py

This will display all available commands and their descriptions for reproducing the complete experimental results from the paper.

File Structure

Main Scripts

• flow_vqe_main.py - Main script for Flow-VQE training and evaluation
• flow_vqe_warm_up.py - Script for generating warm-up parameters using trained Flow-VQE models
• optimization_baselines_main.py - Script for running traditional VQE optimization baselines
• example_usage.py - Comprehensive examples for reproducing all paper results
• ansatz.py - Quantum circuit ansatz definitions and implementations
• molecule_configs.py - Molecular system configurations and parameters
• read_draw.ipynb - Jupyter notebook for data analysis and visualization

Core Packages

• flow_vqe/ - Main Flow-VQE implementation package

  • main.py - Core Flow-VQE training logic
  • flow_training.py - Normalizing flow training procedures
  • molecule_utils.py - Molecular system utilities and setup
  • circuit_utils.py - Quantum circuit manipulation utilities
  • evaluation.py - Model evaluation and metrics
  • plotting.py - Visualization and plotting functions
  • utils.py - General utility functions
  • warm_up.py - Warm-start parameter generation
  • config.py - Configuration management

• optimization_baselines/ - Traditional VQE optimization package

  • runner.py - Optimization algorithm implementations
  • config.py - Baseline optimization configurations
  • plotter.py - Results visualization
  • utils.py - Utility functions for baselines

Data and Results

• a_store_data/ - Experimental results and trained models, find here
  • flow_vqe_m_*/ - Flow-VQE-M training results
  • flow_vqe_s_optimization/ - Flow-VQE-S optimization experiments
  • pt_optimization_results_*/ - Parameter transfer optimization results
  • vqe_optimization_results_*/ - Traditional VQE optimization results
  • warm_optimization_*/ - Warm-start optimization results
  • warm_up/ - Generated warm-up parameters

Contact

If you have any questions or other issues, please contact me at: hangzo@chalmers.se

Cite this paper

@article{zou2025generative,
      title={Generative flow-based warm start of the variational quantum eigensolver}, 
      author={Hang Zou and Martin Rahm and Anton Frisk Kockum and Simon Olsson},
      year={2025},
      eprint={2507.01726},
      archivePrefix={arXiv}
}