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Numerical Feynman integrals with physically inspired interpolation: Faster convergence and significant reduction of computational cost
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26. We assume that the system is coupled to a single reservoir, and that all possible Feynman paths for the system, for the entire duration of the simulation, are included in the Feynman integral. This is to date the most common way Feynman integrals are calculated for ρD in OQSs.
27. Throughout this article, the SI convention will be used rather than the binary convention, so 1TB≡1012 bytes rather than 1TB=1TiB≡240 bytes.
When this article was written, at the top of the Top500.org
list of the most powerful supercomputers, was the Cray XT5 Jaguar which had hard drive space on the order of 10PB.
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One of the most successful methods for calculating reduced density operator dynamics in open quantum systems, that can give numerically exact results, uses Feynman integrals. However, when simulating the dynamics for a given amount of time, the number of time steps that can realistically be used with this method is always limited, therefore one often obtains an approximation of the reduced density operator at a sparse grid of points in time. Instead of relying only on ad hocinterpolation methods (such as splines) to estimate the system density operator in between these points, I propose a method that uses physical information to assist with this interpolation. This method is tested on a physically significant system, on which its use allows important qualitative features of the density operator dynamics to be captured with as little as two time steps in the Feynman integral. This method allows for an enormous reduction in the amount of memory and CPU time required for approximating density operator dynamics within a desired accuracy. Since this method does not change the way the Feynman integral itself is calculated, the value of the density operator approximation at the points in time used to discretize the Feynamn integral will be the same whether or not this method is used, but its approximation in between these points in time is considerably improved by this method. A list of ways in which this proposed method can be further improved is presented in the last section of the article.
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