We present a novel variational quantum framework for partial differential equation (PDE) constrained design optimization problems. Such problems arise in simulation based design in many scientific and engineering domains. For instance in aerodynamic design, the PDE constraints are the conservation laws such as momentum, mass and energy balance, the design variables are vehicle shape parameters and material properties, and the objective could be to minimize the effect of transient heat loads on the vehicle or to maximize the lift. The proposed framework utilizes the variational quantum linear system (VQLS) algorithm and a black box optimizer as its two main building blocks. VQLS is used to solve the linear system, arising from the discretization of the PDE constraints for given design parameters, and evaluate the design cost/objective function. The black box optimizer is used to select next set of parameter values based on this evaluated cost, leading to nested bi-level optimization structure within a hybrid classical-quantum setting. We present detailed complexity analysis to highlight the potential advantages of our proposed framework over classical techniques. We implement our framework using the PennyLane library, apply it to solve a prototypical heat transfer optimization problem, and present simulation results using Bayesian optimization as the black box

翻译：我们提出了一种新颖的变分量子框架，用于解决偏微分方程约束的设计优化问题。此类问题广泛存在于诸多科学与工程领域的基于仿真的设计中。例如在空气动力学设计中，偏微分方程约束为动量、质量和能量平衡等守恒定律，设计变量为飞行器形状参数与材料属性，目标可以是最小化瞬态热载荷对飞行器的影响或最大化升力。所提出的框架以变分量子线性系统算法和黑盒优化器作为两个核心构建模块。VQLS用于求解由给定设计参数下偏微分方程约束离散化所产生的线性系统，并评估设计成本/目标函数。黑盒优化器则根据评估出的成本选择下一组参数值，从而在经典-量子混合环境中形成嵌套的双层优化结构。我们通过详细的复杂度分析，阐明了所提框架相较于经典技术的潜在优势。我们使用PennyLane库实现了该框架，将其应用于解决典型传热优化问题，并给出了以贝叶斯优化作为黑盒优化器的仿真结果。