Iterative solutions to the steady state density matrix for optomechanical systems

TL;DR

This paper introduces a graph theory-based sparse matrix reordering technique that significantly improves the efficiency and stability of iterative solutions for the steady state density matrix in large quantum optomechanical systems.

Contribution

It presents a novel graph-theoretic reordering method that creates stable LU preconditioners, enabling scalable and efficient iterative solutions for complex quantum systems.

Findings

Reduces memory and runtime requirements for large systems

Maintains stability at large Hilbert space dimensions

Enables solutions to previously intractable systems

Abstract

We present a sparse matrix permutation from graph theory that gives stable incomplete Lower-Upper (LU) preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems. This reordering is efficient, adding little overhead to the computation, and results in a marked reduction in both memory and runtime requirements compared to other solution methods, with performance gains increasing with system size. Either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance. This reordering optimizes the condition number of the approximate inverse, and is the only method found to be stable at large Hilbert space dimensions. This allows for steady state solutions to otherwise intractable quantum optomechanical systems.