We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.

Quantum linear system algorithms (QLSAs) have the potential to speed up Interior Point Methods (IPMs). However, a major bottleneck is the inexactness of quantum tomography to extract classical solutions from quantum states. In addition, QLSAs are sensitive to the condition number, and this sensitivity is exacerbated when the Newton systems arising in IPMs converge to a singular matrix. Recently, an Inexact Feasible Quantum IPM (IF-QIPM) has been developed that addresses the inexactness of QLSAs. However, this method requires a large number of gates and qubits to be implemented. Here, we propose a new IF-QIPM using the normal equation system, which requires fewer gates and qubits. To mitigate the sensitivity to the condition number and other input data-related parameters, we use preconditioning coupled with iterative refinement to obtain better complexity. Finally, we demonstrate the effectiveness of our approach on IBM Qiskit simulators.

Quantum computing has the potential to speed up machine learning methods. One major direction is using quantum linear algebra to solve linear system problems or optimization problems in the machine learning area. Quantum approaches in the literature for different types of least squares problems demonstrate speedups w.r.t. dimension but have disadvantages w.r.t. precision and condition number. In this paper, we discuss how an iterative refinement scheme can deliver an accurate solution without the excessive cost of Quantum Linear System Algorithms (QLSAs). In addition, we propose an adaptive regularization approach that can mitigate the effect of condition number on solution time. In the second part of this paper, we investigate how state-of-the-art Quantum Interior Point Methods (QIPMs) can solve more sophisticated regression problems such as Lasso regression and support vector machine problems, which can be reformulated as quadratic optimization problems.