journal article·2026-03-26Open access

Is the Matrix Completion of Reduced Density Matrices Unique?

Gustavo E. Massaccesi, Ofelia B. Oña, Luis Lain, Alicia Torre, Juan E. Peralta, Diego R. Alcoba, Gustavo E. Scuseria

Abstract

High Resolution Image Download MS PowerPoint Slide 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 the computational cost. However, matrix completion is, in general, an under-determined problem. Revisiting Rosina’s theorem ( Rosina, M. Queen’s Papers on Pure and Applied Mathematics, 1968, No. 11, 369), 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.
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Research context

OpenAlex topics: Matrix Theory and Algorithms · Advanced Topics in Algebra · Advanced Mathematical Theories and Applications

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Affiliations

Departamento de Ciencias Exactas, Ciclo Básico Común, Universidad de Buenos Aire, Ciudad Universitaria, 1428 Buenos Aires, Argentina and Instituto de Investigaciones Matemáticas “Luis A. Santaló” (IMAS), Consejo Nacional de Investigaciones Científicas y Técnicas

Universidad de Buenos Aires. Ciudad Universitaria

Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas. Diag.123 y 64 (S/N), Sucursal 4, CC 16, 1900 La Plata, Argentina

Department of Physical Chemistry, Faculty of Science and Technology

University of the Basque Country

Department of Physics

Central Michigan University

Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física. Ciudad Universitaria, 1428 Buenos Aires

Argentina and CONICET - Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA). Ciudad Universitaria

Department of Chemistry

Rice University

Department of Physics and Astronomy

Funding

Welch Foundation · Secretar?a de Ciencia y T?cnica, Universidad de Buenos Aires · Basic Energy Sciences · Consejo Nacional de Investigaciones Cient?ficas y T?cnicas · Agencia Nacional de Promoci?n Cient?fica y Tecnol?gica · Division of Materials Research

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