# Variational Quantum Algorithms for Semidefinite Programming

arXiv: https://arxiv.org/abs/2112.08859 · HTML: https://mlsanity.com/paper/2112.08859/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2021.

## Built on by (most cited first, 8 of 8)

- [Stochastic noise can be helpful for variational quantum algorithms](https://mlsanity.com/paper/2210.06723/) (2022)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2307.01171/) (2023)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization](https://mlsanity.com/paper/2603.04061/) (2026)
- [Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems](https://mlsanity.com/paper/2508.09103/) (2025)
- [Solving Conic Programs over Sparse Graphs using a Variational Quantum Approach: The Case of the AC Optimal Power Flow](https://mlsanity.com/paper/2509.00341/) (2025)
- [Variational Quantum Eigensolver with Constraints (VQEC): Solving Constrained Optimization Problems via VQE](https://mlsanity.com/paper/2311.08502/) (2023)

## Abstract (arXiv)

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., Ngg M, where N is the dimension of the input matrices and M is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.