Is the matrix completion of reduced density matrices unique?

TL;DR

This paper investigates the conditions under which the matrix completion of reduced density matrices in quantum systems is unique and introduces an algorithm for exact reconstruction, enhancing computational efficiency in many-body quantum calculations.

Contribution

The paper proves the uniqueness of matrix completion for 2-RDMs under certain conditions and presents a hybrid quantum-stochastic algorithm for exact reconstruction.

Findings

Matrix completion of 2-RDMs is unique under specific conditions.

A hybrid quantum-stochastic algorithm achieves exact matrix completion.

Applications to the Fermi-Hubbard model demonstrate effectiveness.

Abstract

Reduced density matrices are central to describing observables in many-body quantum systems. In electronic structure theory, the two-particle reduced density matrix (2-RDM) suffices to determine the energy and other key properties. Recent work has used matrix completion, leveraging the low-rank structure of RDMs and approximate theoretical models, to reconstruct the 2-RDM from partial data and thus reduce computational cost. However, matrix completion is, in general, an under-determined problem. Revisiting Rosina's theorem [M. Rosina, Queen's Papers on Pure and Applied Mathematics No. 11, 369 (1968)], we here show that the matrix completion is unique under certain conditions, identifying the subset of 2-RDM elements that enables its exact reconstruction from incomplete information. Building on this, we introduce a hybrid quantum-stochastic algorithm that achieves exact matrix completion, demonstrated through applications to the Fermi-Hubbard model.