In recent years, quantum computing has emerged as a rapidly advancing interdisciplinary field, attracting growing attention from both academia and industry owing to its fundamentally novel approach to information processing and its potential to achieve computational capabilities beyond the reach of classical systems. This paper presents a structured review of quantum computing approaches applied to scheduling problems, which are among the most challenging NP-hard optimization problems. We first introduce fundamental principles of quantum computing, including gate-based models, quantum annealing, and hybrid frameworks. We then explore how various scheduling problems such as job-shop, flow-shop, and project scheduling-are encoded into quantum models, particularly using QUBO formulations suitable for current quantum devices. Finally, we discuss key limitations such as hardware constraints and model scalability, and we identify research directions to advance quantumenhanced scheduling methods. By grounding theory in empirical case studies and highlighting near-term deployable solutions, this study bridges the gap between quantum algorithm development and practical scheduling demands, offering a roadmap for researchers and practitioners in quantum optimization.
In recent years, quantum computing has emerged as a rapidly advancing interdisciplinary field, attracting growing attention from both academia and industry owing to its fundamentally novel approach to information processing and its potential to achieve computational capabilities beyond the reach of classical systems. This paper presents a structured review of quantum computing approaches applied to scheduling problems, which are among the most challenging NP-hard optimization problems. We first introduce fundamental principles of quantum computing, including gate-based models, quantum annealing, and hybrid frameworks. We then explore how various scheduling problems such as job-shop, flow-shop, and project scheduling-are encoded into quantum models, particularly using QUBO formulations suitable for current quantum devices. Finally, we discuss key limitations such as hardware constraints and model scalability, and we identify research directions to advance quantumenhanced scheduling methods. By grounding theory in empirical case studies and highlighting near-term deployable solutions, this study bridges the gap between quantum algorithm development and practical scheduling demands, offering a roadmap for researchers and practitioners in quantum optimization.
Large-scale combinatorial scheduling problems -- job-shop scheduling, flow-shop scheduling, and vehicle routing among them -- remain NP-hard at practically relevant scale, and quantum computing has entered this space along two related tracks: genuinely quantum solvers, principally quantum annealers and gate-based devices running the quantum approximate optimization algorithm (QAOA), and quantum-inspired classical solvers, such as coherent Ising machines and digital annealers, that borrow the same Ising/QUBO problem structure without requiring quantum hardware. This article synthesizes both tracks as applied specifically to scheduling. We trace the theoretical foundations from quantum and simulated annealing through the QUBO formulation techniques that make scheduling problems solver-ready, then review direct applications to job-shop, flow-shop, and vehicle-routing problems, reporting the concrete, verified numbers the underlying studies provide. The consistent finding is that quantum hardware remains constrained by qubit connectivity, embedding overhead, and shallow achievable circuit depth to small benchmark instances, while quantum-inspired classical solvers, unconstrained by qubit count, already reach scheduling-relevant problem sizes with reported performance competitive with or exceeding classical heuristics. We close by identifying benchmarking rigor, hardware connectivity, and the near-total absence of work on dynamic, online scheduling as the field's clearest open problems.
Large-scale combinatorial scheduling problems -- job-shop scheduling, flow-shop scheduling, and vehicle routing among them-- remain NP-hard at practically relevant scale, and quantum computing has entered this space along two related tracks:genuinely quantum solvers, principally quantum annealers and gate-based devices running the quantum approximateoptimization algorithm (QAOA), and quantum-inspired classical solvers, such as coherent Ising machines and digitalannealers, that borrow the same Ising/QUBO problem structure without requiring quantum hardware. This article synthesizesboth tracks as applied specifically to scheduling. We trace the theoretical foundations from quantum and simulated annealingthrough the QUBO formulation techniques that make scheduling problems solver-ready, then review direct applications tojob-shop, flow-shop, and vehicle-routing problems, reporting the concrete, verified numbers the underlying studies provide. Theconsistent finding is that quantum hardware remains constrained by qubit connectivity, embedding overhead, and shallowachievable circuit depth to small benchmark instances, while quantum-inspired classical solvers, unconstrained by qubitcount, already reach scheduling-relevant problem sizes with reported performance competitive with or exceeding classicalheuristics. We close by identifying benchmarking rigor, hardware connectivity, and the near-total absence of work on dynamic,online scheduling as the field's clearest open problems.
Quantum computing has emerged as a transformative paradigm, leveraging principles of quantum mechanics to tackle computationally intractable problems. This manuscript explores the potential of quantum computing for solving complex engineering problems—specifically optimization, simulation, and data analysis tasks—using technologies and methodologies available up to 2022. We examine both gate-based and quantum annealing approaches, outline experimental frameworks, and compare performance against classical counterparts. Results from simulated benchmark problems demonstrate promising speedups in select cases, though practical adoption is constrained by hardware limitations such as qubit coherence times and noise. The study concludes by highlighting avenues for integrating hybrid quantum-classical workflows in engineering, along with a discussion of current constraints and future directions. Over the past decade, research efforts have focused on understanding how quantum phenomena like superposition and entanglement can be harnessed to realize computational advantages for specific problem classes. In particular, combinatorial optimization problems—ubiquitous in engineering design and logistics—offer fertile ground for quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) and quantum annealing. Additionally, quantum simulation techniques, including the Variational Quantum Eigensolver (VQE), promise to accelerate the modeling of physical systems at the molecular and materials level, potentially informing the design of next-generation structural and functional materials. Beyond raw algorithmic performance, this paper carefully assesses the end‑to‑end workflow, accounting for overheads in problem encoding, error mitigation, and classical post‑processing. We also present a detailed analysis of resource utilization—quantified by qubit counts, circuit depths, and annealing schedules—highlighting trade‑offs between solution quality and execution time. Ultimately, our findings underscore that, while pure quantum speedups remain rare in 2022, hybrid approaches harnessing both quantum and classical resources can yield practical benefits for small‑to‑medium scale engineering tasks. This expanded evaluation provides a roadmap for researchers and practitioners interested in applying quantum computing to real‑world engineering challenges.
Recent advancements in quantum computing technology have heralded a new era of computational capabilities, outpacing classical simulations in solving complex problems. This remarkable progress has sparked a burgeoning interest in quantum algorithms across various sectors, with optimization emerging as a particularly vibrant field of study. A focal point of this field is the development and refinement of optimization algorithms that are suitable for current and emerging capability of quantum computers. These advancements are crucial as they offer practical ways to apply quantum computing to real-world challenges. Among these applications, personnel and task execution scheduling is a compelling use case for quantum computing as applied to solve complex optimization problems critical to business and industrial operations. In this paper, we explore the utilization of quantum computers to address this class of optimization problems. Employing a range of advanced techniques, including a novel method that improves efficiency, our method demonstrates the capability of quantum computing to provide effective solutions for intricate optimization problems. We particularly focus on the application of IBM's 127 -qubit quantum devices to manage scheduling tasks in workforce management, involving 500 to 874 binary decision variables and over 1000 constraints. This exploration serves not only as a proof of concept for the practical applications of quantum computing but also as a testament to its evolving role in solving business-critical optimization challenges.
Despite quantum computing is revealing an increasingly promising technology that has the potential to introduce a significant speed-up in many areas of computation, the number of problems that it can represent and solve is currently rather limited. Therefore, one of the current challenges faced by the quantum computing community is to broaden the class of problems that can be tackled. Among these problems, scheduling problems are a class of particularly interesting and hard combinatorial problems; in this paper, we present a novel solution for representing and solving the Flexible Open Shop Scheduling Problem (FOSSP) to optimality by minimizing the makespan. We firstly present a compact formulation of this problem as a Quadratic unconstrained binary optimization (QUBO), which can be used to solve this problem with a quantum annealer. Then, we proceed to the Quantum Approximate Optimization Algorithm (QAOA) problem formulation, thus producing both the cost and mix Hamiltonians related to the problem. From the Hamiltonians, we provide the complete description of the quantum circuit that can be used to tackle the FOSSP within the QAOA framework. This second approach can be used to solve the optimization problem with a general-purpose quantum gate-based hardware.
Quantum computing is rapidly evolving as a paradigm that promises to solve complex computational problems far beyond the capabilities of classical systems. This article explores the current challenges facing quantum computation, particularly in algorithm development, error correction, and hardware scalability. The paper also outlines the prospects of next-generation quantum algorithms, especially those applicable to cryptography, optimization, and machine learning. Recent advances in quantum error mitigation and hybrid quantum-classical models are reviewed, emphasizing the transition from theoretical constructs to practical deployment. This study aims to bridge the gap between quantum algorithm research and its implementation, offering insights into the interdisciplinary efforts needed to make quantum advantage a reality.