Geometric Optimization on Lie Groups: A Lie-Theoretic Explanation of Barren Plateau Mitigation for Variational Quantum Algorithms

This project provides a Lie-theoretic framework for analyzing and mitigating barren plateaus in Variational Quantum Algorithms (VQAs).

We model parameter evolution as trajectories on the Lie group, and study optimization dynamics using geometric quantities such as:

• Velocity
• Directional acceleration
• Path length and endpoint distance ratio
• Stability metrics (relative standard deviation)

Project Structure

We have scripts

.
├── lie_params_anaylze.py # analyze the parameters
├── model_training.py # training the quantum circuit and record the parameters
└── run_lie_analyze.sh

Shell script for running the full pipeline

• Trains quantum circuits
• Records parameter
• Executes Lie-geometric analysis
• Generates plots and metrics

And core package modules

.
├── lie_analyze
│   ├── data_record
│   │   ├── __init__.py
│   │   └── param_record.py # record all the parameters
│   ├── model
│   │   ├── __init__.py
│   │   └── model.py # nueral network we used
│   ├── preparation
│   │   ├── __init__.py
│   │   ├── config.py #global configuration 
│   │   ├── cost_function.py
│   │   ├── gpu_info.py
│   │   ├── quantum_circuit.py
│   │   └── random_state.py # random input quantum state
│   ├── __init__.py
└── lib.rs

Environment Setup

This project uses uv for dependency management.

Install uv (if not already installed)

curl -LsSf https://astral.sh/uv/install.sh | sh

# if you don't have curl
wget -qO- https://astral.sh/uv/install.sh | sh

Clone this repo

git clone https://github.com/rahulbhadani/cpe487587hw01
cd LIE_ANALYZE
uv venv --python 3.12
source .venv/bin/activate
uv sync
uv build

Using the script

Make it executable

chmod +x run_lie_analyze.sh

Running the Full Pipeline

bash scripts/run_lie_analyze.sh

Output

For each qubit configuration, the output will under

Parameter_Analysis/qXX/Analysis_*/

Citation

@article{yi2025geometric,
  title={Geometric Optimization on Lie Groups: A Lie-Theoretic Explanation of Barren Plateau Mitigation for Variational Quantum Algorithms},
  author={Yi, Zhehao and Bhadani, Rahul},
  journal={arXiv preprint arXiv:2512.02078},
  year={2025}
}