Conic programs arise broadly in physics, quantum information, machine learning, and engineering, many of which are defined over sparse graphs. Although such problems can be solved in polynomial time using classical interior-point solvers, the computational complexity scales unfavorably with graph size. In this context, this work proposes a variational quantum paradigm for solving conic programs, including quadratically constrained quadratic programs (QCQPs) and semidefinite programs (SDPs). We encode primal variables via the state of a parameterized quantum circuit (PQC), and dual variables via the probability mass function of a second PQC. The Lagrangian function can thus be expressed as scaled expectations of quantum observables. A primal-dual solution can be found by minimizing/maximizing the Lagrangian over the parameters of the first/second PQC. We pursue saddle points of the Lagrangian in a hybrid fashion. Gradients of the Lagrangian are estimated using the two PQCs, while PQC parameters are updated classically using a primal-dual method. We propose permuting the primal variables so that related observables are expressed in a banded form, enabling efficient measurement. The proposed framework is applied to the OPF problem, a large-scale optimization problem central to the operation of electric power systems. Numerical tests on the IEEE 57-node power system using Pennylane's simulator corroborate that the proposed doubly variational quantum framework can find high-quality OPF solutions. Although showcased for the OPF, this framework features a broader scope, including conic programs with numerous variables and constraints, problems defined over sparse graphs, and training quantum machine learning models to satisfy constraints.

翻译：锥规划问题广泛出现于物理学、量子信息、机器学习及工程学领域，其中许多问题定义在稀疏图上。尽管此类问题可使用经典内点法求解器在多项式时间内求解，但其计算复杂度随图规模增长而呈现不利的缩放特性。在此背景下，本研究提出一种用于求解锥规划问题的变分量子范式，包括二次约束二次规划（QCQP）和半定规划（SDP）。我们通过参数化量子电路（PQC）的量子态对原始变量进行编码，并通过第二个PQC的概率质量函数对偶变量进行编码。拉格朗日函数因此可表达为量子可观测量缩放期望值的形式。原始-对偶解可通过在第一个/第二个PQC的参数上最小化/最大化拉格朗日函数求得。我们采用混合方式寻找拉格朗日函数的鞍点：利用两个PQC估计拉格朗日函数的梯度，同时使用原始-对偶方法经典地更新PQC参数。我们提出对原始变量进行置换，使相关观测量呈现带状形式，从而实现高效测量。所提出的框架被应用于最优潮流（OPF）问题——电力系统运行中的核心大规模优化问题。在IEEE 57节点电力系统上使用Pennylane模拟器进行的数值测试证实，所提出的双重变分量子框架能够找到高质量的最优潮流解。尽管以最优潮流问题为展示案例，该框架具有更广泛的应用范围，包括具有大量变量和约束的锥规划问题、定义在稀疏图上的问题，以及训练量子机器学习模型以满足约束条件。