Geometric Shadow Inversion (GSI)

Real-Time Quantum State Tomography via Kernel Regression

Abstract

Quantum state tomography (QST) is the bottleneck for characterizing intermediate-scale quantum devices. Standard Maximum Likelihood Estimation (MLE) becomes computationally intractable and statistically unstable when data is sparse. Geometric Shadow Inversion (GSI) is a reconstruction framework that treats quantum state recovery as kernel regression on the density matrix manifold.

By utilizing a Gaussian-weighted adjoint projection, GSI triangulates state geometry in a single non-iterative pass. We identify an Information Threshold at measurement ratio $M/P \approx 1.17$, above which GSI enables robust, high-fidelity reconstruction ($F > 0.99$) independent of bandwidth tuning.

Paper: Moralez, M. (2025). Geometric Shadow Inversion: Real-Time Quantum State Tomography at the Information Threshold (1.0.0). Zenodo. https://doi.org/10.5281/zenodo.17716380

Interactive visualization https://gsi-research-visualization-850185611796.us-west1.run.app/

Features

• Fast: Reconstructs 5-qubit states in $\approx 10$ ms and 6-qubit states in $\approx 5$ min (CPU).

• Robust: Operates in the under-determined regime ($M/P < 1$) where linear inversion fails.

• Physical: Guarantees positive semi-definite (PSD) output via geometric projection.

• No-MLE: Eliminates iterative optimization and convergence issues.

Installation

1. Clone the repository:

   git clone [https://github.com/MarcosMoralez/gsi-tomography.git](https://github.com/MarcosMoralez/gsi-tomography.git)
   cd gsi-tomography
   

2. Install dependencies:

   pip install -r requirements.txt
   

Usage

import numpy as np
from gsi_core import GeometricShadowTomography, generate_random_mixed_state

# 1. Setup System
qubits = 4
engine = GeometricShadowTomography(qubits, sigma=0.78)

# 2. Generate Truth & Simulate Measurements
true_rho = generate_random_mixed_state(2**qubits)
data = engine.measure_state(true_rho, num_settings=500, shots=1000)

# 3. Reconstruct (Single Pass)
rho_est = engine.reconstruct(data)

# 4. Validate
fidelity = engine.fidelity(true_rho, rho_est)
print(f"Reconstruction Fidelity: {fidelity:.4f}")

Reproducing Paper Results

To reproduce the scaling benchmark (Table I in the paper) for 3, 4, 5, and 6 qubits:

python benchmarks/reproduce_paper_results.py

To generate "The MLE Trap" visualization (Figure 1):

python figures/plot_mle_trap.py

Citation

If you use this code in your research, please cite our work:

@article{Moralez2025GSI,
  title={Geometric Shadow Inversion: Real-Time Quantum State Tomography at the Information Threshold},
  author={Moralez, Marcos T.},
  year={2025},
  publisher={Zenodo},
  doi={10.5281/zenodo.17716380},
  url={https://doi.org/10.5281/zenodo.17716380}
}

License

This project is licensed under the MIT License - see the [LICENSE] file for details.

Note: This software is the subject of a pending patent application.