We present a novel variational quantum framework for linear partial differential equation (PDE) constrained optimization problems. Such problems arise in many scientific and engineering domains. For instance, in aerodynamics, 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-to-drag ratio. 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 computational error and 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 a heat transfer optimization problem, and present simulation results using Bayesian optimization as the black box optimizer.

翻译：我们提出了一种新颖的变分量子框架，用于解决线性偏微分方程（PDE）约束优化问题。这类问题出现在众多科学与工程领域。例如，在空气动力学中，PDE约束包括动量、质量和能量守恒等守恒定律，设计变量为飞行器形状参数和材料属性，优化目标可以是最小化瞬态热载荷对飞行器的影响，或最大化升阻比。该框架以变分量子线性系统（VQLS）算法和黑盒优化器作为两大核心构建模块。VQLS用于求解给定设计参数下由PDE约束离散化产生的线性系统，并评估设计成本/目标函数；黑盒优化器则基于评估后的成本选择下一组参数值，从而在混合经典-量子设置中形成嵌套的双层优化结构。我们通过详细的计算误差与复杂度分析，揭示了该框架相较于经典技术的潜在优势。我们利用PennyLane库实现了该框架，并将其应用于热传递优化问题，同时展示了采用贝叶斯优化作为黑盒优化器的仿真结果。