Quantum circuit optimization plays a critical role in improving the performance of algorithms executed on noisy intermediate-scale quantum (NISQ) devices. While modern quantum compilers provide multiple optimization levels, selecting an appropriate optimization strategy remains a non-Trivial task due to the complex interplay between circuit structure and hardware noise. In this work, we present a comprehensive noise-Aware evaluation of quantum circuit optimization strategies using a diverse set of 33 circuits, including Quantum Fourier Transform (QFT), Grover's search, Variational Quantum Eigensolver (VQE), and randomly generated circuits. We analyze the impact of different transpiler optimization levels on circuit depth, entanglement (CNOT count), and output fidelity under a realistic depolarizing noise model. Our experimental results reveal that no single optimization level consistently yields the best performance across all circuit types. While aggressive optimization is beneficial for structured circuits such as QFT and VQE, moderate optimization levels often provide a better trade-off between circuit complexity and fidelity for unstructured or randomly generated circuits. These findings highlight that optimization-level selection is inherently dependent on circuit characteristics and cannot be effectively addressed using static or heuristic approaches alone. To address this challenge, we further propose a lightweight learning-based model that predicts the optimal optimization level based on circuit features. The model achieves an accuracy of 93.9% across the evaluated dataset, demonstrating that simple machine learning techniques can effectively approximate optimal compiler decisions without requiring exhaustive evaluation. Overall, this work provides practical insights into optimization behavior in NISQ systems and introduces a scalable approach for adaptive optimization selection, contributing toward more efficient quantum software development.

Within the framework of quantum information theory, the performance of conventional 2-D techniques is often unsatisfactory for the study of Bell states that hold a unique status as maximally entangled states. Therefore, 3-D approaches are increasingly employed to achieve more accurate and detailed analysis, offering improved performance and insights in complex scenarios. Among the diverse background of mixed two-qubit states, certain configurations exhibit unique quantum correlations that harness advanced quantum information processing tasks such as Bell diagonal states. Although, these states may appear superficially simple and exhibit a rich spectrum of correlations. The present research employs a methodology that involves a convex combination of distinct Bell states to generate the entire class of Bell diagonal states. This work explores the time evolution of Bell diagonal states, when exposed to various quantum channels and investigates the dynamics of quantum correlations such as entanglement, discord, and state of separability. Finally, the behaviour of Bell diagonal states is analysed and results are compared between theory and practice. A threedimensional visual approach is used to illustrate a deeper understanding of various quantum features and dynamic behaviour of the Bell diagonal states.

Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed "NP-completeness"as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow-Hassidim-Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm's potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.

Bell states are special sets of entangled states that serve as a foundation for the corresponding Hilbert space. Being the maximally entangled states, they have significant importance in the field of quantum information theory. Among the mixed two-qubit state space, a highly intriguing limited class of states is the Bell diagonal states. Despite their inherent simplicity, Bell diagonal states exhibits a diverse range of correlations. In this paper, we explore the behaviour of Bell diagonal states under different quantum channels to pursue the geometrical illustration of quantum correlations leading to the entanglement. We propose the convex combination of distinct Bell states to prepare the Bell diagonal states. The whole class of Bell diagonal states is generated and entanglement is simulated throughout each parameter space and compared to the theory.