# Constrained shadow tomography for molecular simulation on quantum devices

## TL;DR

This paper introduces a new method for reconstructing quantum states from limited data, improving accuracy and scalability for quantum simulations.

## Contribution

A bi-objective semidefinite programming approach for constrained shadow tomography with N-representability and regularization.

## Key findings

- The method improves accuracy and noise resilience in 2-RDM reconstruction.
- Numerical and hardware results show enhanced scalability for quantum simulations.
- Physical consistency is enforced through N-representability constraints.

## Abstract

Quantum state tomography is a fundamental task in quantum information science, enabling detailed characterization of correlations, entanglement, and electronic structure in quantum systems. However, its exponential measurement and computational demands limit scalability, motivating efficient alternatives such as classical shadows, which enable accurate prediction of many observables from randomized measurements. In this work, we introduce a bi-objective semidefinite programming approach for constrained shadow tomography, designed to reconstruct the two-particle reduced density matrix (2-RDM) from noisy or incomplete shadow data. By integrating N-representability constraints and nuclear-norm regularization into the optimization, the method builds an N-representable 2-RDM that balances fidelity to the shadow measurements with energy minimization. This unified framework mitigates noise and sampling errors while enforcing physical consistency in the reconstructed states. Numerical and hardware results demonstrate that the approach significantly improves accuracy, noise resilience, and scalability, providing a robust foundation for physically consistent fermionic state reconstruction in realistic quantum simulations.

Quantum state tomography is a fundamental task in quantum information science, enabling detailed characterization of correlations, entanglement, and electronic structure in quantum systems.