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We provide the necessary framework for carrying out stochastic positive-P andgauge-P simulations of bosonic systems with long range interactions. In theseapproaches, the quantum evolution is sampled by trajectories in phase space,allowing calculation of correlations without truncation of the Hilbert space orother approximations to the quantum state. The main drawback is that thesimulation time is limited by noise arising from interactions.  We show that the long-range character of these interactions does not furtherincrease the limitations of these methods, in contrast to the situation foralternatives such as the density matrix renormalisation group. Furthermore,stochastic gauge techniques can also successfully extend simulation times inthe long-range-interaction case, by making using of parameters that affect thenoise properties of trajectories, without affecting physical observables.  We derive essential results that significantly aid the use of these methods:estimates of the available simulation time, optimized stochastic gauges, ageneral form of the characteristic stochastic variance and adaptations for verylarge systems. Testing the performance of particular drift and diffusion gaugesfor nonlocal interactions, we find that, for small to medium systems, driftgauges are beneficial, whereas for sufficiently large systems, it is optimal touse only a diffusion gauge.  The methods are illustrated with direct numerical simulations of interactionquenches in extended Bose-Hubbard lattice systems and the excitation of Rydbergstates in a Bose-Einstein condensate, also without the need for the typicalfrozen gas approximation. We demonstrate that gauges can indeed lengthen theuseful simulation time.

Quantum dynamics of long-range interacting systems using the positive-P and gauge-P representations