Localized density matrix minimization and linear scaling algorithms

TL;DR

This paper introduces a convex variational method with  regularization to compute localized density matrices efficiently, enabling linear scaling algorithms for quantum systems at both zero and finite temperatures.

Contribution

It presents a novel  regularized variational approach that guarantees convergence and produces localized density matrices with banded structure for linear computational complexity.

Findings

The method effectively approximates quantum systems with exponential decay of density matrices.

The algorithms achieve linear computational cost relative to problem size.

Theoretical analysis confirms the approximation quality and convergence guarantees.

Abstract

We propose a convex variational approach to compute localized density matrices for both zero temperature and finite temperature cases, by adding an entry-wise $\ell_1$ regularization to the free energy of the quantum system. Based on the fact that the density matrix decays exponential away from the diagonal for insulating system or system at finite temperature, the proposed $\ell_1$ regularized variational method provides a nice way to approximate the original quantum system. We provide theoretical analysis of the approximation behavior and also design convergence guaranteed numerical algorithms based on Bregman iteration. More importantly, the $\ell_1$ regularized system naturally leads to localized density matrices with banded structure, which enables us to develop approximating algorithms to find the localized density matrices with computation cost linearly dependent on the problem size.