Universal Dilation of Linear It\^o SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments

TL;DR

This paper introduces a universal quantum simulation framework for linear Itô SDEs using unitary dilation, enabling efficient trajectory and moment simulations on quantum computers with rigorous error analysis.

Contribution

It develops a novel unitary dilation method to embed classical linear SDEs into quantum stochastic Schrödinger equations, facilitating quantum simulation of second moments.

Findings

Pathwise exact embedding of classical solutions into quantum states.

Efficient quantum algorithms for trajectory and moment simulations.

Validated error bounds and numerical experiments.

Abstract

We present a universal framework for simulating $N$-dimensional linear It\^o 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 \[ dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j \] to stochastic Schr\"odinger 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.