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Iterative solutions to the steady state density matrix for optomechanical systems
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.Comment: 10 pages, 5 figure
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