Computational framework for quantum state tomography of spin ensembles

Quantum state tomography (QST) is essential for characterizing quantum systems, yet the practical procedures for transforming raw spectroscopic data into density matrices are often poorly documented or hidden within proprietary software. This paper addresses the need for a transparent, open-source computational framework that allows researchers to perform, validate, and reproduce QST analysis without relying on closed-source commercial tools.

The authors developed Tomography-NMR, a Python package that implements a complete analysis pipeline based on the product operator formalism. The workflow processes raw time-domain NMR signals through Fourier transformation, phase correction, and peak integration to extract the coefficients of the density matrix. The package offers three integration methods: direct peak height measurement, fixed-parameter numerical integration (suitable for unknown states), and a systematic grid-search optimization (for benchmarking known states). The software is modular, allowing users to adapt data import routines to different spectrometer formats.

Tomography-NMR was validated on 20 experimentally prepared two-qubit states, including computational basis states, Bell states, and outputs of fundamental quantum gates (CNOT, H, and T). The software achieved average reconstruction fidelities between 0.975 and 0.995. The optimized integration method consistently outperformed simpler approaches, demonstrating that systematic parameter selection is critical for achieving high-fidelity results. The package also includes robust error propagation and visualization tools, making it a comprehensive solution for NMR-based quantum information experiments.

By exposing every step of the data processing pipeline—from spectral preprocessing to density matrix visualization—this package fills a significant gap in the literature. It enables independent verification of experimental results, provides a standardized tool for comparing quantum characterization across research groups, and offers a low-barrier entry point for students and new practitioners in the field of quantum information processing.