Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments

Abstract: 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 correspondence between the general linear SDE [ dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j ] and a Stochastic Schrödinger Equation (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact: the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSE is {naturally implementable} on digital quantum processors, where the stochastic Wiener increments correspond directly to measurement outcomes of 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 simulations.