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arXiv · 2302.01962

Correspondence between open bosonic systems and stochastic differential equations

Alexander Engel·Scott E. Parker
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Abstract

Bosonic mean-field theories can approximate the dynamics of systems of $n$ bosons provided that $n \gg 1$. We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment and the mean-field theory is replaced by a stochastic differential equation. When the $n \to \infty$ limit is taken, the stochastic terms in this differential equation vanish, and a mean-field theory is recovered. Besides providing insight into the differences between the behavior of finite quantum systems and their classical limits given by $n \to \infty$, the developed mathematics can provide a basis for quantum algorithms that solve some stochastic nonlinear differential equations. We discuss conditions on the efficiency of these quantum algorithms, with a focus on the possibility for the complexity to be polynomial in the log of the stochastic system size. A particular system with the form of a stochastic discrete nonlinear Schr\"{o}dinger equation is analyzed in more detail.

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Alexander Engel, Scott E. Parker. 2023-02-03. Correspondence between open bosonic systems and stochastic differential equations. https://doi.org/10.1140/epjp/s13360-023-04205-9

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