This paper presents a comprehensive review and systematic coverage of foundational quantum optimization algorithms-QAOA, Quantum Annealing, VQE, VQLS, and HHL-tracing their development all the way from Schrödinger's equation through gate-based quantum computing (Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, QFT, Shor's, and Grover's algorithms) to hybrid quantum-classical approaches for NISQ-era hardware. We also present the transverse field Ising model's role in quantum annealing as well as practical hardware implementations with the Chimera, Pegasus, and Zephyr topologies of the D-Wave quantum annealer, addressing key challenges including barren plateaus, circuit trainability, and noise resilience. In the simulations section, quantum computing applications to collaborative filtering and social network optimization have been presented. We introduce two original contributions: (1) a quantum collaborative filtering model based on VQE principles that leverages quantum entanglement to represent user-item relationships, requiring logarithmically fewer parameters than classical embedding methods while accessing exponentially larger state spaces, and demonstrating test RMSE of 0.85 on synthetic data; and (2) a Quantum Social Network (QSN) framework modeling belief dynamics through superposition and entanglement, revealing emergent consensus and polarization phenomena validated via Qiskit simulations. By demonstrating that variational quantum algorithms can tackle real-world recommendation and network optimization problems on current hardware, we bridge theoretical quantum computing with practical machine learning applications and identify concrete pathways toward quantum advantage beyond traditional combinatorial optimization.
This paper presents a comprehensive review and systematic coverage of foundational quantum optimization algorithms-QAOA, Quantum Annealing, VQE, VQLS, and HHL-tracing their development all the way from Schrödinger's equation through gate-based quantum computing (Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, QFT, Shor's, and Grover's algorithms) to hybrid quantum-classical approaches for NISQ-era hardware. We also present the transverse field Ising model's role in quantum annealing as well as practical hardware implementations with the Chimera, Pegasus, and Zephyr topologies of the D-Wave quantum annealer, addressing key challenges including barren plateaus, circuit trainability, and noise resilience. In the simulations section, quantum computing applications to collaborative filtering and social network optimization have been presented. We introduce two original contributions: (1) a quantum collaborative filtering model based on VQE principles that leverages quantum entanglement to represent user-item relationships, requiring logarithmically fewer parameters than classical embedding methods while accessing exponentially larger state spaces, and demonstrating test RMSE of 0.85 on synthetic data; and (2) a Quantum Social Network (QSN) framework modeling belief dynamics through superposition and entanglement, revealing emergent consensus and polarization phenomena validated via Qiskit simulations. By demonstrating that variational quantum algorithms can tackle real-world recommendation and network optimization problems on current hardware, we bridge theoretical quantum computing with practical machine learning applications and identify concrete pathways toward quantum advantage beyond traditional combinatorial optimization.
Quantum optimization allows for up to exponential quantum speedups for specific, possibly industrially relevant problems. As the key algorithm in this field, we motivate and discuss the Quantum Approximate Optimization Algorithm (QAOA), which can be understood as a slightly generalized version of Quantum Annealing for gate-based quantum computers. We delve into the quantum circuit implementation of the QAOA, including Hamiltonian simulation techniques for higher-order Ising models, and discuss parameter training using the parameter shift rule. An example implementation with Pennylane source code demonstrates practical application for the Maximum Cut problem. Further, we show how constraints can be incorporated into the QAOA using Grover mixers, allowing to restrict the search space to strictly valid solutions for specific problems. Finally, we outline the Variational Quantum Eigensolver (VQE) as a generalization of the QAOA, highlighting its potential in the NISQ era and addressing challenges such as barren plateaus and ansatz design.
The convergence of machine learning (ML) and quantum computing is quickly becoming a game-changing new frontier in computational science. The most intriguing area of overlap may be quantum optimization, which seeks to reduce the exponentially increasing complexity and computational cost of ML algorithms. As they grow larger and more complicated, traditional optimization methods, particularly in high-dimensional and non-convex space, will inevitably hit performance limits. Quantum optimization is the whole new paradigm that leverages quantum mechanical effects like superposition, entanglement, and tunnelling to explore solution spaces more efficiently. The chapter outlines the theory, practice, and potential of hybrid quantum-classical optimization algorithms for machine learning. The most brilliant research direction in the current NISQ (noisy intermediate-scale quantum) era is mixed systems leveraging classical computing capability augmented by quantum capacity. In such designs, computationally intensive subroutines like optimization for global optima or escape from local minima are left to quantum processors and data preprocessing, iterative refinement, and model testing are performed by classical processors. We start by defining the fundamental principles of quantum computing and acquainting readers with the fundamentals like qubits, quantum gates, and quantum circuits. These are the building blocks for quantum algorithms, which hopefully can be utilized for optimization, for example, the QAOA (quantum approximate optimization algorithm), the VQE (variational quantum Eigensolver), and quantum annealing. They all build to optimize the problems in some way differently: QAOA through parameterized quantum circuits, VQE through approximations of ground states of the Hamiltonians, and quantum annealing through quantum evolution to low-energy solutions. We then form a loop between these quantum algorithms and regular machine learning tasks. Optimization impacts virtually all ML phases, from training neural networks, where one reduces loss functions; feature selection, where one aims to discover the most relevant subset of input variables; and hyperparameter optimization, which often involves expensive search procedures over extremely large combinatorial spaces. Quantum optimization techniques can therefore, in principle, accelerate such procedures through computational complexity reduction and enhancing solutions over challenging landscapes. Realistic simulations and case studies emphasize the degree to which hybrid quantum-classical algorithms have been employed in solving ML problems, that is, QSVMs, quantum-enhanced clustering, and quantum reinforcement learning. Despite their experimental nature, these applications already show encouraging results in terms of improved performance metrics as well as reduction in time-to-solution compared to classical-only solutions. Notwithstanding that, this new field is not bereft of real challenges. Quantum hardware suffers from hard limits on qubit fidelity, decoherence, and scalability in the current times. Furthermore, quantum circuit integration into broad ML pipelines requires novel architectures and software platforms. As a counter to those issues, this chapter surveys recent advances in quantum programming languages, simulators, and hybrid platforms like PennyLane, Qiskit, and TensorFlow Quantum. We conclude by looking to the future possibility that quantum optimization will revolutionize the machine learning landscape. As hardware improves and algorithmic complexity increases, the possibility of quantum-powered learning systems solving hard problems becomes increasingly accessible. From speeding up deep network training to enabling new types of unsupervised learning, quantum optimization holds the potential to recast computational tractability in machine learning. This chapter is an all-encompassing introduction and state-of-the-art work for students, researchers, and practitioners interested in using quantum optimization for practical ML applications. It sets the foundation for understanding how to pursue hybrid approaches (how they may be applied today in the real world and bracing oneself for disruption in the future).
Quantum annealing has emerged as a transformative approach in solving combinatorial optimization problems by leveraging the principles of quantum mechanics. A comprehensive survey of quantum annealing is presented, delineating its foundational concepts, state-of-the-art advancements, and diverse applications across domains such as logistics, finance, machine learning, and energy optimization. Notable contributions include the introduction of a systematic benchmarking framework to evaluate quantum annealing against classical optimization techniques across metrics like time-to-target, scalability, and energy efficiency. Additionally, novel hybrid quantum-classical paradigms and hardware advancements are explored, addressing challenges in qubit connectivity, coherence, and noise resilience. Optimization problems are mapped to suitable quantum annealing platforms, offering practical guidelines for leveraging quantum technologies in real-world scenarios. Critical gaps in scalability and integration with machine learning are identified, with future research directions proposed to bridge these challenges. These findings provide a holistic understanding of quantum annealing's potential, offering a foundational resource for advancing quantum computing in optimization.
This chapter investigates the application of quantum computing to optimization, a field traditionally dominated by classical methods but increasingly limited by computational demands in tackling complex, large-scale problems. The discussion focuses on quantum optimization techniques, particularly variational quantum algorithms like the Quantum Approximate Optimization Algorithm (QAOA) and the variational quantum eigensolver (VQE), along with quantum annealing (QA), a process inspired by physical annealing that leverages quantum tunnelling and superposition. The chapter also explores the integration of quantum approaches within machine learning frameworks, where quantum algorithms can potentially accelerate model training and enhance capabilities. Additionally, it presents practical applications and challenges associated with these quantum algorithms, covering their effectiveness on noisy intermediate-scale quantum (NISQ) devices and the evolving landscape of quantum hardware. This chapter concludes with insights into quantum optimization's future potential to address optimization and machine learning challenges that are currently intractable for classical methods, marking a significant shift in computational science.
This review critically examines the preprint by Chakrabarti, Changdar, and Khanda (2026), which provides a comprehensive survey of quantum computing algorithms applied to mathematical optimization. The paper traces the evolution from foundational quantum mechanics to hybrid quantum-classical methods, while introducing novel contributions in quantum collaborative filtering and social network optimization. We contextualize the survey within the broader quantum computing literature, highlighting its pedagogical strengths in bridging theory and application. Key postulates, such as the role of superposition and entanglement in optimization, are evaluated alongside major findings, including simulations demonstrating practical feasibility on noisy intermediate-scale quantum (NISQ) hardware. Comparisons with contemporaneous works reveal the survey's unique emphasis on variational algorithms for machine learning tasks, though gaps in empirical validation persist. We conclude with directions for future research, advocating for scalable implementations and interdisciplinary integrations. This review underscores the paper's value as an accessible entry point for optimization researchers entering quantum paradigms. Also available on Academia.edu: https://www.academia.edu/attachments/131787157/download_file?s=portfolio
Combinatorial optimization is a foundation of numerous scientific, industrial, and social decision-making questions, but its usefulness is mostly restricted by the exponential complexity of the problem of classical computability. The quantum annealing (QA) has become one of the leading analogue quantum systems to address such problems, whereby optimization problems are converted into physical energy structures and quantum fluctuations are utilized to perform systematic exploration. It provides a critical, synthesized, and comprehensive review of quantum annealing as applied to combinatorial optimization, including the theoretical basis, hardware designs, algorithm methods, issues of embedding and encoding, benchmarking procedures, applications, and integration with quantum algorithms based on gates and solvers on a classical computer. We build a comprehensive taxonomy between adiabatic dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to state-of-the-art flux-qubit annealers, new architectures, and hybrid quantum classical pipelines. We show that the overhead of embedding and encoding and the benchmarking approach taken are the most significant factors in determining scalability and performance, but not the raw number of qubits. Transportation, energy systems, robotics, finance, pharmacological discovery, and machine learning are just some examples of the domain-specific case studies that use QA with minimal empirical utility as a hybrid refinement engine and not an independent solver. Moreover, we introduce a stringent evaluation of the existing benchmarking culture, clarify the causal links between QA and QAOA as well as between VQE and structural predicaments, and reveal the institutional roots of the existing restrictions. We conclude by identifying a research roadmap in the future, focusing on the annealing hardness characterization, Stoquastic control manipulation, embedding rules automation, architecture design, and development of principled quantum advantage. It creates a benchmark reference to those in the field and research and deployment of scalable, reliable, and application-relevant quantum optimization