Is the Matrix Completion of Reduced Density Matrices Unique?

Area of Science:

• Quantum Many-Body Physics
• Computational Chemistry
• Quantum Information Theory

Background:

• Reduced density matrices (RDMs) are crucial for describing quantum systems.
• The two-particle reduced density matrix (2-RDM) is sufficient for calculating electronic structure properties.
• Matrix completion methods reconstruct RDMs from partial data, reducing computational expense.

Purpose of the Study:

• To determine conditions for unique and exact reconstruction of the 2-RDM.
• To develop a computational method for exact 2-RDM matrix completion.
• To apply the method to challenging quantum systems like the Fermi-Hubbard model.

Main Methods:

• Revisiting and applying Rosina's theorem to identify unique 2-RDM reconstruction conditions.
• Developing a hybrid quantum-stochastic algorithm for exact matrix completion.
• Utilizing the low-rank structure of RDMs and approximate theoretical models.

Main Results:

• Identified a specific subset of 2-RDM elements that guarantees unique reconstruction.
• Demonstrated exact matrix completion using the novel hybrid algorithm.
• Successfully applied the method to the Fermi-Hubbard model, validating its efficacy.

Conclusions:

• The matrix completion of 2-RDMs can be exact and unique under specific conditions.
• The developed hybrid quantum-stochastic algorithm provides an efficient route to exact 2-RDM reconstruction.
• This approach offers significant potential for reducing computational costs in electronic structure calculations and quantum simulations.