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

• URL: http://arxiv.org/abs/2601.05928v1
• Date: Fri, 09 Jan 2026 16:42:25 GMT
• Title: Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments
• Authors: Hsuan-Cheng Wu, Xiantao Li,
• Abstract summary: We present a universal framework for simulating linear It differential equations (SDEs) on quantum computers with additive or multiplicative noises.<n>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=1J B_j(t)X_t,dW_tj ] and a Schrdinger Equation (SSE) on a dilated Hilbert space.
• Score: 2.847280100380157
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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