Correspondence between open bosonic systems and stochastic differential equations

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 Schrödinger equation is analyzed in more detail.

50 pages, 0 figures

References in corpus (10)

• Quantum algorithm for solving linear systems of equations
• Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques
• Creating superpositions that correspond to efficiently integrable probability distributions
• Efficient quantum algorithm for dissipative nonlinear differential equations
• Many-Body Quantum Spin Dynamics with Monte Carlo Trajectories on a Discrete Phase Space
• Solving nonlinear differential equations with differentiable quantum circuits
• Simulating spin-boson dynamics with stochastic Liouville-von Neumann equations
• Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms
• Beating the House: Fast Simulation of Dissipative Quantum Systems with Ensemble Rank Truncation
• Open quantum dynamics of strongly coupled oscillators with multi-configuration time-dependent Hartree propagation and Markovian quantum jumps

Cited by in corpus (1)

• Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems