The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands to capture the uncertainty arising from the sparse function evaluations. Quantum computers have emerged as potential replacements for classical computers, offering exponential reductions in the computational complexity of machine learning tasks. In this paper, we combine Gaussian process quadratures and quantum computing by proposing a quantum low-rank Gaussian process quadrature method based on a Hilbert space approximation of the Gaussian process kernel and enhancing the quadrature using a quantum circuit. The method combines the quantum phase estimation algorithm with the quantum principal component analysis technique to extract information up to a desired rank. Then, Hadamard and SW AP tests are implemented to find the expected value and variance that determines the quadrature. We use numerical simulations of a quantum computer to demonstrate the effectiveness of the method. Furthermore, we provide a theoretical complexity analysis that shows a polynomial advantage over classical Gaussian process quadrature methods.

Reinforcement Learning (RL) has become a critical tool for optimization challenges within automation, leading to significant advancements in several areas. This review article examines the current landscape of RL within automation, with a particular focus on its roles in manufacturing, energy systems, and robotics. It discusses state-of-the-art methods, major challenges, and upcoming avenues of research within each sector, highlighting RL's capacity to solve intricate optimization challenges. The paper reviews the advantages and constraints of RL-driven optimization methods in automation. It points out prevalent challenges encountered in RL optimization, including issues related to sample efficiency and scalability; safety and robustness; interpretability and trustworthiness; transfer learning and meta-learning; and real-world deployment and integration. It further explores prospective strategies and future research pathways to navigate these challenges. Additionally, the survey includes a comprehensive list of relevant research papers, making it an indispensable guide for scholars and practitioners keen on exploring this domain.

Gaussian processes are probabilistic models that are commonly used as functional priors in machine learning. Due to their probabilistic nature, they can be used to capture the prior information on the statistics of noise, smoothness of the functions, and training data uncertainty. However, their computational complexity quickly becomes intractable as the size of the data set grows. We propose a Hilbert space approximation-based quantum algorithm for Gaussian process regression to overcome this limitation. Our method consists of a combination of classical basis function expansion with quantum computing techniques of quantum principal component analysis, conditional rotations, and Hadamard and Swap tests. The quantum principal component analysis is used to estimate the eigenvalues while the conditional rotations and the Hadamard and Swap tests are employed to evaluate the posterior mean and variance of the Gaussian process. Our method provides polynomial computational complexity reduction over the classical method.