A primal-dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

TL;DR

This paper introduces a specialized primal-dual semidefinite programming algorithm designed for efficiently solving the variational two-body density matrix problem in quantum many-body physics, demonstrating good performance on pairing Hamiltonians.

Contribution

A tailored primal-dual semidefinite programming algorithm that exploits problem structure for efficient computation in quantum density matrix determination.

Findings

Efficient matrix-vector product computation.

Standard N-representability conditions perform well.

Algorithm successfully applied to pairing Hamiltonian.

Abstract

The quantum many-body problem can be rephrased as a variational determination of the two-body reduced density matrix, subject to a set of N-representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a standard primal-dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard N-representability conditions perform very well for this problem.