2015/04/25 by Paul D. Nation, P. D. Nation, Nation, P. D. · 2 citations
Computer Science · Physics and Astronomy · #Computational Physics (physics.comp-ph) #FOS: Physical sciences #Matrix Theory and Algorithms #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph) #physics.comp-ph #quant-ph
paper · pdf · doi:10.48550/arxiv.1504.06768
21 pages, 4 figures. arXiv admin note: text overlap with arXiv:1411.4356
arxiv created 2015/04/25 · openalex publication_date 2015/04/25 · arxiv updated 2015/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss the numerical solution methods available when solving for the steady-state density matrix of a time-independent open quantum optical system, where the system operators are expressed in a suitable basis representation as sparse matrices. In particular, we focus on the difficulties posed by the non-Hermitian structure of the Lindblad super operator, and the numerical techniques designed to mitigate these pitfalls. In addition, we introduce a doubly iterative inverse-power method that can give reduced memory and runtime requirements in situations where other iterative methods are limited due to poor bandwidth and profile reduction. The relevant methods are demonstrated on several prototypical quantum optical systems where it is found that iterative methods based on iLU factorization using reverse Cuthill-Mckee ordering tend to outperform other solution techniques in terms of both memory consumption and runtime as the size of the underlying Hilbert space increases. For eigenvalue solving, Krylov iterations using the stabilized bi-conjugate gradient method outperform generalized minimal residual methods. In contrast, minimal residual methods work best for solvers based on direct LU decomposition. This work serves as a guide for solving the steady-state density matrix of an arbitrary quantum optical system, and points to several avenues of future research that will extend the applicability of these classical algorithms in absence of a quantum computer.