arXiv subjects

Pouya Sampourmahani

Publications and source records attributed to Pouya Sampourmahani.

5 recordsLinked to original sources

arXiv2025

Quantum Interior Point Methods: A Review of Developments and An Optimally Scaling Framework

The growing demand for solving large-scale, data-intensive linear and conic optimization problems, particularly in applications such as artificial intelligence and machine learning, has highlighted the limitations of classical interior point methods (IPMs). Despite their favorable polynomial-time convergence, conventional IPMs often suffer from high per-iteration computational costs, especially for dense problem instances. Recent advances in quantum computing, particularly quantum linear system solvers, offer promising avenues to accelerate the most computationally intensive steps of IPMs. However, practical challenges such as quantum error, hardware noise, and sensitivity to poorly conditioned systems remain significant obstacles. In response, a series of Quantum IPMs (QIPMs) has been developed to address these challenges, incorporating techniques such as feasibility maintenance, iterative refinement, and preconditioning. In this work, we review this line of research with a focus on our recent contributions, including an almost-exact QIPM framework. This hybrid quantum-classical approach constructs and solves the Newton system entirely on a quantum computer, while performing solution updates classically. Crucially, all matrix-vector operations are executed on quantum hardware, enabling the method to achieve an optimal worst-case scalability w.r.t dimension, surpassing the scalability of existing classical and quantum IPMs.

quant-ph↗

arXiv2025

Optimal Scaling Quantum Interior Point Method for Linear Optimization

The emergence of huge-scale, data-intensive linear optimization (LO) problems in applications such as machine learning has driven the need for more computationally efficient interior point methods (IPMs). While conventional IPMs are polynomial-time algorithms with rapid convergence, their per-iteration cost can be prohibitively high for dense large-scale LO problems. Quantum linear system solvers have shown potential in accelerating the solution of linear systems arising in IPMs. In this work, we introduce a novel almost-exact quantum IPM, where the Newton system is constructed and solved on a quantum computer, while solution updates occur on a classical machine. Additionally, all matrix-vector products are performed on the quantum hardware. This hybrid quantum-classical framework achieves an optimal worst-case scaling of $\mathcal{O}(n^2)$ for fully dense LO problems. To ensure high precision, despite the limited accuracy of quantum operations, we incorporate iterative refinement techniques both within and outside the proposed IPM iterations. The proposed algorithm has a quantum complexity of $\mathcal{O}(n^{1.5} κ_A \log(\frac{1}ε))$ queries to QRAM and $\mathcal{O}(n^2 \log(\frac{1}ε))$ classical arithmetic operations. Our method outperforms the worst-case complexity of prior classical and quantum IPMs, offering a significant improvement in scalability and computational efficiency.

quant-ph↗

arXiv2024

A quantum dual logarithmic barrier method for linear optimization

Quantum computing has the potential to speed up some optimization methods. One can use quantum computers to solve linear systems via Quantum Linear System Algorithms (QLSAs). QLSAs can be used as a subroutine for algorithms that require solving linear systems, such as the dual logarithmic barrier method (DLBM) for solving linear optimization (LO) problems. In this paper, we use a QLSA to solve the linear systems arising in each iteration of the DLBM. To use the QLSA in a hybrid setting, we read out quantum states via a tomography procedure which introduces considerable error and noise. Thus, this paper first proposes an inexact-feasible variant of DLBM for LO problems and then extends it to a quantum version. Our quantum approach has quadratic convergence toward the central path with inexact directions and we show that this method has the best-known $\mathcal{O}(\sqrt{n} \log (n μ_0 /ζ))$ iteration complexity, where $n$ is the number of variables, $μ_0$ is the initial duality gap, and $ζ$ is the desired accuracy. We further use iterative refinement to improve the time complexity dependence on accuracy. For LO problems with quadratically more constraints than variables, the quantum complexity of our method has a sublinear dependence on dimension.

math.OC↗

arXiv2023

Quantum Computing Inspired Iterative Refinement for Semidefinite Optimization

Iterative Refinement (IR) is a classical computing technique for obtaining highly precise solutions to linear systems of equations, as well as linear optimization problems. In this paper, motivated by the limited precision of quantum solvers, we develop the first IR scheme for solving semidefinite optimization (SDO) problems and explore two major impacts of the proposed IR scheme. First, we prove that the proposed IR scheme exhibits quadratic convergence toward an optimal solution without any assumption on problem characteristics. We also show that using IR with Quantum Interior Point Methods (QIPMs) leads to exponential improvements in the worst-case overall running time of QIPMs, compared to previous best-performing QIPMs. We also discuss how the proposed IR scheme can be used with classical inexact SDO solvers, such as classical inexact IPMs with conjugate gradient methods.

math.OC↗

arXiv2023

On Semidefinite Representations of Second-order Conic Optimization Problems

Second-order conic optimization (SOCO) can be considered as a special case of semidefinite optimization (SDO). In the literature it has been advised that a SOCO problem can be embedded in an SDO problem using the arrow-head matrix transformation. However, a primal-dual solution pair cannot be mapped simultaneously using the arrow-head transformation as we might lose complementarity and duality in some cases. To address this issue, we investigate the relationship between SOCO problems, and their SDO counterpart. Through derivation of standard semidefinite representations of SOCO problems, we introduce admissible mappings. We show that the proposed mappings preserve both feasibility and optimality. Further, we discuss how the optimal partition of a SOCO problem maps to the optimal partition of its SDO counterpart.

math.OC↗