Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments
- 1Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
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 mapping from the general linear SDEs d X t = A ( t ) X t d t + ∑ j = 1 J B j ( t ) X t d W t j 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.
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BibTeX
@article{wu2026universal,
title = {{Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments}},
author = {Hsuan-Cheng Wu and Xiantao Li},
journal = {Quantum},
year = {2026},
month = oct,
doi = {10.22331/q-2026-10-01-2223},
url = {https://quantum-journal.org/papers/q-2026-10-01-2223/},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments
AU - Hsuan-Cheng Wu
AU - Xiantao Li
PY - 2026
DA - 2026/10/01
JO - Quantum
DO - 10.22331/q-2026-10-01-2223
UR - https://quantum-journal.org/papers/q-2026-10-01-2223/
AB - 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 d X t = A ( t ) X t d t + ∑ j = 1 J B j ( t ) X t d W t j 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.
SN - 2521-327X
ER -