Representation of stochastic nonlinear dynamical systems with open bosonic quantum systems

One approach for approximating nonlinear dynamics on quantum computers is to express the target dynamical system as a mean-field theory for a quantum system of many identical particles which can be simulated efficiently with quantum algorithms. However, rigorous analyses indicate that maintaining accuracy of bosonic mean-field systems out to long times t requires a number of bosons n that is exponentially large in t. Therefore, that approach does not appear to allow for a quantum speedup of nonlinear dynamics beyond short simulation times. Instead, we generalize this idea to open bosonic quantum systems, and find that there is an exact mapping to stochastic nonlinear dynamical systems. The stochastic terms give finite-n corrections to the mean-field theory evolution. We demonstrate this mapping through numerical testing of simple systems. If the target dynamics is actually given by a stochastic dynamical system, or if weak stochastic terms are acceptable as errors to the target dynamics, then a fixed number of bosons n can be considered for arbitrary t. This may allow for a quantum speedup based on quantum simulation of the open bosonic system.