# Quantum Interior Point Methods for Semidefinite Optimization

arXiv: https://arxiv.org/abs/2112.06025 · HTML: https://mlsanity.com/paper/2112.06025/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2021.

## Built on by (most cited first, 6 of 6)

- [Quantum Langevin Dynamics for Optimization](https://mlsanity.com/paper/2311.15587/) (2023)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Quantum Power Flows: From Theory to Practice](https://mlsanity.com/paper/2211.05728/) (2022)
- [Quantum Algorithms for Bandits with Knapsacks with Improved Regret and Time Complexities](https://mlsanity.com/paper/2507.04438/) (2025)
- [Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems](https://mlsanity.com/paper/2508.09103/) (2025)
- [Differentiable Quantum Computing for Large-scale Linear Control](https://mlsanity.com/paper/2411.01391/) (2024)

## Abstract (arXiv)

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem n, but has worse dependence on other numerical parameters.