Solving Conic Programs over Sparse Graphs using a Variational Quantum Approach: The Case of the Optimal Power Flow

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.