Autonomous Mobile Robots (AMRs) play an increasingly important role in flexible and reconfigurable production systems. Assigning heterogeneous tasks to multiple AMRs constitutes a combinatorial optimization problem that becomes computationally intractable as system complexity grows. This paper presents a Quadratic Unconstrained Binary Optimization (QUBO) model for AMR task assignment, integrating scheduling and resource allocation within a Hamiltonian formulation. The model is parameterized and evaluated using both exact and heuristic solvers to analyze scalability and validity across benchmark scenarios. Results demonstrate the trade-off between computational effort and solution optimality, confirming the feasibility of the QUBO representation for this problem class. The formulated model creates a basis for future implementation on quantum hardware for optimization using Quantum Annealing (QA) or the Quantum Approximate Optimization Algorithm (QAOA), thereby bridging the gap between classical optimization methods and new quantum computing paradigms.
Autonomous Mobile Robots (AMRs) play an increasingly important role in flexible and reconfigurable production systems. Assigning heterogeneous tasks to multiple AMRs constitutes a combinatorial optimization problem that becomes computationally intractable as system complexity grows. This paper presents a Quadratic Unconstrained Binary Optimization (QUBO) model for AMR task assignment, integrating scheduling and resource allocation within a Hamiltonian formulation. The model is parameterized and evaluated using both exact and heuristic solvers to analyze scalability and validity across benchmark scenarios. Results demonstrate the trade-off between computational effort and solution optimality, confirming the feasibility of the QUBO representation for this problem class. The formulated model creates a basis for future implementation on quantum hardware for optimization using Quantum Annealing (QA) or the Quantum Approximate Optimization Algorithm (QAOA), thereby bridging the gap between classical optimization methods and new quantum computing paradigms.
Autonomous Mobile Robots (AMRs) are increasingly deployed in smart factories and intralogistics systems, yet dynamic fleet scheduling remains challenging because of growing task complexity, multi-objective requirements, and human–robot interaction. This review examines the evolution of AMR scheduling from Mixed-Integer Linear Programming (MILP), heuristics, metaheuristics, and Reinforcement Learning (RL) toward emerging quantum and hybrid quantum–classical optimization. Particular attention is given to Quadratic Unconstrained Binary Optimization (QUBO), the Quantum Approximate Optimization Algorithm (QAOA), and quantum annealing as potential approaches for representing coupled task allocation, routing, energy management, and human-safety requirements. The industrial implications of quantum-enhanced scheduling are critically discussed in terms of operational coordination, resource efficiency, human-centric collaboration, and scalability. Sector-specific opportunities and current research maturity are further assessed from an Industry 5.0 perspective. The review identifies hybrid quantum–classical architectures, dynamic QUBO formulations, Digital Twin integration, and human-aware scheduling as key research directions for advancing scalable and adaptive multi-AMR fleet coordination.
Quantum annealing (QA) offers a physics-inspired approach for such subproblems by mapping them onto an Ising Hamiltonian, which is algebraically equivalent to a Quadratic Unconstrained Binary Optimization (QUBO) model executable on quantum annealers or hybrid solvers. However, their practical employment in wireless systems critically depends on the specific problem formulation, on the particular variable encoding, constraint handling, solver selection, hardware embedding, and fair benchmarking. This paper develops wireless-oriented QUBO modeling guidelines for QA-assisted resource allocation. We first present a beginner-friendly toy example in which a terrestrial-aerial resource-allocation problem with a fractional signal-to-interference-plus-noise ratio (SINR) maximization objective is reformulated into a two-bit QUBO via Boolean interpolation, converted into an Ising representation, and decoded back into a wireless scheduling decision. The same scheduling is applicable to UAV, aircraft, satellite, and terrestrial links in SAGIN systems. We then review the QA foundations and solver ecosystems, including binary quadratic, discrete quadratic, constrained quadratic, and hybrid models from the perspective of NG wireless optimization. Furthermore, we discuss both binary and discrete encoding, penalty-based constraint conversion, embedding overhead, solution decoding, and feasibility verification. A QUBO Compatibility Index is introduced as a lightweight diagnostic metric for assessing the associated QUBO formulation readiness, rather than a predictor of runtime or solution quality. Finally, a case study on max-min uplink power control in cell-free massive MIMO illustrates how a fairness-oriented wireless problem can be discretized, constrained, converted into the concept of a QUBO, and evaluated against classical baselines. A host of key challenges and future directions are also discussed.
Combinatorial Optimization (CO) is one of the most important areas in the field of optimization, with practical applications found in every industry, including both the private and public sectors. In recent years, it was discovered that a mathematical formulation known as QUBO (Quadratic Unconstrained Binary Optimization) problem can embrace an exceptional variety of important CO problems found in the industry. In this work, we explore the Quantum Approximate Optimization Algorithm (QAOA) as a quantum computing approach to solve the Knapsack Problem efficiently by formulating the problem as a QUBO model. Here we implemented QAOA on a quantum emulator, leveraging quantum superposition and entanglement to explore multiple solutions simultaneously. The method offers a promising route for solving large, complex scheduling and resource allocation problems that are tough for classical algorithms, with potential applications in logistics, HPC resource scheduling, and energy optimization.
Combinatorial optimization problems are typically formulated using Quadratic Unconstrained Binary Optimization (QUBO), where constraints are enforced through penalty terms that introduce auxiliary variables and rapidly increase Hamiltonian complexity, limiting scalability on near term quantum devices. In this work, we systematically study Quadratic Unconstrained D-ary Optimization (QUDO) as an alternative formulation in which decision variables are encoded directly in higher dimensional Hilbert spaces. We demonstrate that QUDO naturally captures structural constraints across a range of problem classes, including the Traveling Salesman Problem, two variants of the Vehicle Routing Problem, graph coloring, job scheduling, and Max-K-Cut, without the need for extensive penalty constructions. Using a qudit-level implementation of the Quantum Approximate Optimization Algorithm (qudit QAOA), we benchmark these formulations against their binary QUBO counterparts and exact classical solutions. Our study show consistently improved approximation ratios and substantially reduced computational overhead at comparable circuit depths, highlighting QUDO as a scalable and expressive representation for quantum combinatorial optimization.
Quantum optimization which is the use of quantum computers and algorithms for solving complex optimization problems, is one of the topics in quantum computing with the highest potential. Two types of approaches are generally used to solve combinatorial optimization problems with quantum computers: exact methods such as the Grover’s search algorithm, and meta-heuristics such as the Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA). Exact methods and variational methods like QAOA make use of universal gate-based quantum computers, such as the ones developed by the IBM company.Quantum annealing is designed for analogic quantum computers developed by D-Wave. Solving a combinatorial optimization problem with heuristics generally requires a transformation of the problem to a format suitable for the quantum computer. While some existing frameworks can cope with Constrained Quadratic Models (CQM), Quadratic Unconstrained Binary Optimization (QUBO) is the best option to map an optimization problem to a quantum computer or simulator. We consider in this paper the job shop scheduling problem.
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.