Statistical Methods for Quantum State Tomography: Evaluating Maximum Likelihood Estimation and Variational Autoencoders

A systematic comparison of Maximum Likelihood Estimation (MLE) and Variational Autoencoders (VAE) for Quantum State Tomography (QST), evaluated on GHZ, W, and product states across 3-qubit systems – scaling up to 8 qubits to expose MLE's computational limits and assess VAE as a practical, scalable alternative.

Final project for the course Statistics and Data Analysis 2025–2026
M.Sc. in Physics — University of Milano-Bicocca

Authors:

• Simone Razzetti — GitHub
• Riccardo Ruggeri — GitHub

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Overview

Quantum State Tomography is a fundamental protocol in the NISQ era, used for gate benchmarking, noise mitigation verification, and fidelity estimation of prepared states. Classical reconstruction methods suffer from the curse of dimensionality: the Hilbert space grows exponentially with the number of qubits, making full density matrix reconstruction computationally prohibitive.

This project implements and compares two approaches:

• MLE: Maximum Likelihood Estimation via Cholesky parametrization, optimized with iminuit (Migrad algorithm)
• VAE: Variational Autoencoder – via keras package – trained on POVM measurement outcomes to learn the Born probability distribution directly, bypassing explicit density matrix reconstruction

Both methods are evaluated using Classical Fidelity (Bhattacharyya coefficient) as the primary metric, enabling a fair direct comparison without requiring full state reconstruction.

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Project Structure

statistical-qst-vae/
├── data/               # CSV files with simulation results
├── docs/               # Project report (PDF)
├── figs/               # Generated figures and plots
├── notebooks/
│   ├── 01_demo_GHZ3.ipynb      # General demo on proposed methods
│   ├── 02_MLE_analysis.ipynb   # MLE fidelities and scaling analysis
│   ├── 03_VAE_analysis.ipynb   # VAE performance and scalabing analysis
│   └── 04_comparative_analysis.ipynb  # MLE vs VAE comparison
├── src/                # Source modules (data generation, MLE, VAE, metrics)
├── requirements.txt
├── LICENSE
└── README.md

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Installation

Clone the repository and install dependencies:

git clone https://github.com/srazzetti/statistical-qst-vae.git
cd statistical-qst-vae
pip install -r requirements.txt

Or install core dependencies manually:

pip install qiskit==2.4.1 qiskit-aer==0.17.2 tensorflow==2.21.0 keras==3.14.1 \
            numpy==2.4.6 pandas==3.0.3 matplotlib==3.11.0 scipy==1.17.1 \
            seaborn==0.13.2 iminuit==2.32.0 scikit-learn==1.9.0

Tested on Python 3.10+

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References

Key references for this project:

• Chen et al. (2021) — Reconstructing a quantum state with a variational autoencoder, Int. J. Quantum Inf. 19(08):2140005
• Nielsen & Chuang — Quantum Computation and Quantum Information, Cambridge University Press

A complete bibliography is available in the project report (docs/).

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License

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