Transforming the Lindblad Equation into a System of Linear Equations: Performance Optimization and Parallelization
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Rapidly growing performance and memory capacity of modern supercomputers open new perspectives for numerical studies of open quantum systems. These systems are in contact with their environments and their modeling is typically based on Markovian kinetic equations, describing the evolution of the system density operators. Additional to the exponential growth of the system dimension N with the number of the system's parts, the computational complexity scales quadratically with N, since we have to deal now with super-operators represented by N^2 × N^2 matrices (instead of standard N × N matrices of operators). In this paper we consider the so-called Lindblad equation, a popular tool to model dynamics of open systems in quantum optics and superconducting quantum physics. Using the generalized Gell-Mann matrices as a basis, we transform the original Lindblad equation into a system of ordinary differential equations (ODEs) with real coefficients. Earlier, we proposed an implementation of this idea with the complexity of computations scaling as O(N^5 log(N)) for dense systems and O(N^3 log(N)) for sparse systems. However, infeasible memory costs remained an obstacle on the way to large models. Here we present a new parallel algorithm based on cluster manipulations with large amount of data needed to transform the original Lindblad equation into an ODE system. We demonstrate that the algorithm allows us to integrate sparse systems of dimension N=2000 and dense systems of dimension N=200 by using up to 25 nodes with 64 GB RAM per node. We also managed to perform large-scale supercomputer sampling to study the spectral properties of an ensemble of random Lindbladians for N=200, which is impossible without the distribution of memory costs among cluster nodes.
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