We present a universal framework for simulating
N-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous mapping from the general linear SDEs
dXt=A(t)Xtdt+∑j=1JBj(t)XtdWtj to stochastic Schrödinger equations (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact in that the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSEs are {naturally implementable} on digital quantum processors, where the stochastic Wiener increments are encoded directly by preparing the ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical experiments.
We show that well above threshold the use of linearized Ito quantum stochastic differential equations allows a systematic description of all quantum stochastic properties of laser light including the phase properties. This is accomplished by the use of mixed field-atom operators which are phase independent and around whose finite mean values we may linearize. As a consequence we are able to derive an equation for an operator describing the phase properties of the laser light. We illustrate our method by applying it to three different models for sub-shot-noise lasers.