Correspondence between open bosonic systems and stochastic differential equations

Bosonic mean-field theories can approximate the dynamics of systems of n bosons provided that n >> 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 -> infinity 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 -> infinity, 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 Schrodinger equation is analyzed in more detail.

Automated summary

Bosonic mean-field theories can accurately represent systems with a large number of bosons, and a similar accurate representation can be achieved at finite n by including interactions with the environment and using a stochastic differential equation. As n approaches infinity, the stochastic terms vanish and the mean-field theory is recovered. This mathematical development not only provides insights into the differences between finite quantum systems and their classical limits, it also lays the foundation for quantum algorithms solving stochastic nonlinear differential equations. The efficiency of these quantum algorithms is discussed, with a focus on the possibility of polynomial complexity in the log of the stochastic system size. A specific system, the stochastic discrete nonlinear Schrodinger equation, is analyzed in detail.