Solving conic programs over sparse graphs using a variational quantum approach: The case of the AC optimal power flow
Phys. Rev. A 114, 042413 – Published 8 October, 2026
DOI: https://doi.org/10.1103/wp8q-lc5k
Abstract
Conic programs arising in physics, quantum information, machine learning, and engineering are oftentimes defined over sparse graphs. Although such problems can be solved in polynomial time using classical interior-point solvers, the computational complexity scales unfavorably with the graph size. In this context, this work proposes a variational quantum paradigm for solving conic programs, including quadratically constrained quadratic programs and semidefinite programs. We encode primal variables via the state of a parametrized quantum circuit (PQC) and dual variables via the probability mass function associated with a second PQC. The Lagrangian function can thus be expressed as scaled expectations of quantum observables. We pursue approximately stationary points of the Lagrangian by minimizing (maximizing) the Lagrangian over the parameters of the first (second) PQC. This is accomplished 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. We provide a complexity analysis that is useful to determine which problem types may enjoy quantum advantage. The proposed framework is applied to the ac optimal power flow (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. While this OPF demonstration does not yield a quantum speedup, the results serve as a proof of concept and highlight challenges toward practical quantum advantage. Although showcased for the OPF, this framework has a broader scope, including conic programs with many variables and constraints, problems defined over sparse graphs, and training quantum machine-learning models to satisfy constraints.
