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

arXiv: https://arxiv.org/abs/2509.00341 · HTML: https://mlsanity.com/paper/2509.00341/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 74 references that are in the map.
- First posted on arXiv 30 Aug 2025.

## Builds on (most cited first, 24 of 24)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Semidefinite relaxations for certifying robustness to adversarial examples](https://mlsanity.com/paper/1811.01057/) (2018)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization](https://mlsanity.com/paper/2011.00364/) (2020)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms](https://mlsanity.com/paper/1908.02788/) (2019)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [Variational Quantum Algorithms for Semidefinite Programming](https://mlsanity.com/paper/2112.08859/) (2021)
- [PSSE Redux: Convex Relaxation, Decentralized, Robust, and Dynamic Approaches](https://mlsanity.com/paper/1708.03981/) (2017)
- [Noisy intermediate-scale quantum algorithm for semidefinite programming](https://mlsanity.com/paper/2106.03891/) (2021)
- [Quantum Computing for Power Flow Algorithms: Testing on real Quantum Computers](https://mlsanity.com/paper/2204.14028/) (2022)
- [Quantum Power Flow](https://mlsanity.com/paper/2104.04888/) (2021)
- [A Quantum Approach for Stochastic Constrained Binary Optimization](https://mlsanity.com/paper/2301.01443/) (2023)
- [Quantum Power Flows: From Theory to Practice](https://mlsanity.com/paper/2211.05728/) (2022)
- [Equivalent Sufficient Conditions for Global Optimality of Quadratically Constrained Quadratic Program](https://mlsanity.com/paper/2303.05874/) (2023)
- [Limitations of Fault-Tolerant Quantum Linear System Solvers for Quantum Power Flow](https://mlsanity.com/paper/2402.08617/) (2024)
- [Advancing Hybrid Quantum Neural Network for Alternative Current Optimal Power Flow](https://mlsanity.com/paper/2410.20275/) (2024)

## Built on by (most cited first, 1 of 1)

- [Learning AC Power Flow Solutions using a Data-Dependent Variational Quantum Circuit](https://mlsanity.com/paper/2509.03495/) (2025)

## Abstract (arXiv)

Conic programs arising in physics, quantum information, machine learning, and engineering are often 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. We propose 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 parameterized 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 are estimated using the two PQCs, while their parameters are updated classically using a primal-dual method. We propose permuting primal variables so that related observables have a banded form, enabling efficient measurement. We provide a complexity analysis that is useful to determine which problem types may enjoy quantum advantage. The framework is applied to the AC OPF problem, a large-scale optimization problem central to electric power system operation. Numerical tests on the IEEE 57-node system using PennyLane's simulator show that the proposed doubly variational quantum framework can find high-quality OPF solutions. While this 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 OPF, the framework has broader scope, including conic programs with many variables and constraints, problems defined over sparse graphs, and training quantum machine learning models to satisfy constraints.