Simulation of quantum stochastic differential equation

Franco Fagnola

Jincheng Zhang

This paper introduces Quantum-Inspired Stochastic Differential Equations (QSDEs), a novel approach to modeling stochastic systems by leveraging core concepts from quantum mechanics. Traditional Stochastic Differential Equations (SDEs) often rely on Gaussian noise, which can be a significant limitation when accurately representing complex, real-world phenomena. QSDEs propose a fundamental shift by incorporating superposition and entanglement principles to model noise, moving beyond Gaussian distributions. The core mechanism involves introducing a "quantum noise" term, denoted as *η(t, x, P)*, where *η* is a superposition of probability distributions dynamically adjusted based on the system's state *x*. This allows for a more nuanced representation of uncertainty, potentially leading to vastly improved simulations across diverse fields. We outline the theoretical framework, explore the key mathematical formulations, and discuss the potential impact of QSDEs on areas such as financial modeling, physics simulations, and control systems. The central claim of this work is that classical SDEs can be significantly enhanced by incorporating quantum mechanical principles.

Jincheng Zhang

Quantum-Inspired Stochastic Differential Equation (Q-SDE) Optimization addresses the challenges of optimizing complex, stochastic systems by integrating quantum-inspired stochastic differential equation modeling. This research presents a novel optimization algorithm, the Q-SDE-based method, designed to effectively handle highly non-linear, stochastic optimization problems characterized by significant noise. We develop a framework that utilizes the inherent stochasticity of Q-SDEs, coupled with differential equation principles, to provide a robust and adaptive approach to finding optimal solutions. The core mechanism involves a dynamic adjustment of the Q-SDE's parameters to minimize the error function across a wider range of solutions, mitigating the impact of noise and enhancing the algorithm's performance in challenging scenarios. This work demonstrates the effectiveness of this approach through extensive numerical simulations and comparisons with established optimization techniques, highlighting its potential for significant improvements in the optimization of stochastic systems.

Xiangyu Li, Ahmet Burak Catli, Ho Kiat Lim +4 more

Nonlinear stochastic differential equations (NSDEs) are a pillar of mathematical modeling for scientific and engineering applications. Accurate and efficient simulation of large-scale NSDEs is prohibitive on classical computers due to the large number of degrees of freedom, and it is challenging on quantum computers due to the linear and unitary nature of quantum mechanics. We develop a quantum algorithm to tackle nonlinear differential equations driven by the Ornstein-Uhlenbeck (OU) stochastic process. The query complexity of our algorithm scales logarithmically with the error tolerance and nearly quadratically with the simulation time. Our algorithmic framework comprises probabilistic Carleman linearization (PCL) to tackle nonlinearity coupled with stochasticity, and stochastic linear combination of Hamiltonian simulations (SLCHS) to simulate stochastic non-unitary dynamics. We obtain probabilistic exponential convergence for the Carleman linearization of Liu et al. [1], provided the NSDE is stable and reaches a steady state. We extend deterministic LCHS to stochastic linear differential equations, retaining near-optimal parameter scaling from An et al. [2] except for the nearly quadratic time scaling. This is achieved by using Monte Carlo integration for time discretization of both the stochastic inhomogeneous term in LCHS and the truncated Dyson series for each Hamiltonian simulation.

J. Martin Lindsay, Adam Skalski

Mads Peter Soerensen