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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

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.

Is the matrix completion of reduced density matrices unique?

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.
Paper Structure (1 section, 1 theorem, 6 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 1 section, 1 theorem, 6 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Acknowledgments

Key Result

Theorem 1

("Uniqueness RDM Completion Theorem") For an $N$-particle Hamiltonian with at most two-particle interactions with reduced representation $\{\mathrm{^2H}_{ij;kl}\}$ and a non-degenerate ground state, the subset of elements of the 2-RDM corresponding to the ground state associated with the non-zero el

Figures (4)

  • Figure 1: Heat-map representations in the lattice-site basis of the target 2-RDM (left panel), and the reconstructed (fully evolved) density-matrix elements obtained via matrix completion (right panel) corresponding to the non-degenerate ground state of the inhomogeneous three-site one-dimensional Fermi–Hubbard model with open boundary conditions at half-filling. Density-matrix elements associated with the symmetrized non-zero (zero) elements of the corresponding two-particle reduced Hamiltonian are shown as the lower (upper) triangular part of the matrices.
  • Figure 2: Heat-map representations in the eigenbasis of the two-particle reduced Hamiltonian of the target 2-RDM (left panel), and the reconstructed (fully evolved) density-matrix elements obtained via matrix completion (right panel) corresponding to the non-degenerate ground state of the inhomogeneous three-site one-dimensional Fermi–Hubbard model with open boundary conditions at half-filling. Density-matrix elements associated with the symmetrized non-zero (zero) elements of the corresponding two-particle reduced Hamiltonian are shown as the lower (upper) triangular part of the matrices.
  • Figure 3: Left panel: Partial (critical subset of elements) and complete (full set of elements) Hilbert–Schmidt distances between the reduced two-particle state of the unitarily evolved $N$-particle density matrix, $^2\Gamma$, and the target 2-RDM, $^2\Gamma_{\text{t}}$, respectively, as a function of iteration number $n$, for the exact ground state of the model system in Figure \ref{['fig1']}. The evolution starts from an arbitrary linear combination of the ground and first-excited states. Inset: enlarged view of the first 200 iterations. Right panel: Energy deviation and infidelity of the evolved $N$-particle density matrix with respect to the exact ground-state density matrix as a function of iteration number $n$.
  • Figure : Hybrid quantum–stochastic reduced density matrix completion algorithm

Theorems & Definitions (2)

  • Theorem 1
  • proof