Abstract Combinatorial optimization problems are considered to be an application, where quantum computing can have transformative impact. In the industrial context, job shop scheduling problems that aim at finding the optimal schedule for a set of jobs to be run on a set of machines are of immense interest. Here we introduce an efficient encoding of job shop scheduling problems, which requires much fewer bit-strings for counting all possible schedules than previously employed encodings. For problems consisting of N jobs with N operations, the number of required bit-strings is at least reduced by a factor N / log 2 ( N ) as compared to time indexed encodings. This is particularly beneficial for solving job shop scheduling problems on quantum computers, since much fewer qubits are needed to represent the problem. Our approach applies to the large class of flexible and usual job-shop scheduling problems, where operations can possibly be executed on multiple machines. Using variational quantum algorithms, we show that the encoding we introduce leads to significantly better performance of quantum algorithms than previously considered strategies. Importantly, the encoding we develop also enables significantly more compact classical representations and will therefore be highly useful even beyond applicability on quantum hardware.
Abstract Combinatorial optimization problems are considered to be an application, where quantum computing can have transformative impact. In the industrial context, job shop scheduling problems that aim at finding the optimal schedule for a set of jobs to be run on a set of machines are of immense interest. Here we introduce an efficient encoding of job shop scheduling problems, which requires much fewer bit-strings for counting all possible schedules than previously employed encodings. For problems consisting of N jobs with N operations, the number of required bit-strings is at least reduced by a factor N / log 2 ( N ) as compared to time indexed encodings. This is particularly beneficial for solving job shop scheduling problems on quantum computers, since much fewer qubits are needed to represent the problem. Our approach applies to the large class of flexible and usual job-shop scheduling problems, where operations can possibly be executed on multiple machines. Using variational quantum algorithms, we show that the encoding we introduce leads to significantly better performance of quantum algorithms than previously considered strategies. Importantly, the encoding we develop also enables significantly more compact classical representations and will therefore be highly useful even beyond applicability on quantum hardware.
Flexible job-shop scheduling problems (FJSPs) represent one of the most complex combinatorial optimization challenges. Modern production systems and control processes demand rapid decision-making in scheduling. To address this challenge, we propose a quantum computing approach for solving FJSPs. We propose a quadratic unconstrained binary optimization (QUBO) model to minimize the makespan of FJSPs, with the scheduling scheme encoded in the ground state of the Hamiltonian operator. The model is solved using a coherent Ising machine (CIM). Numerical experiments are conducted to evaluate and validate the performance and effectiveness of the CIM. The results demonstrate that quantum computing holds significant potential for solving FJSPs more efficiently than traditional computational methods.
Abstract Combinatorial optimization models a vast range of industrial processes aiming at improving their efficiency. In general, solving this type of problem exactly is computationally intractable. Therefore, practitioners rely on heuristic solution approaches. Variational quantum algorithms are optimization heuristics that can be demonstrated with available quantum hardware. In this case study, we apply four variational quantum heuristics running on IBM’s superconducting quantum processors to the job shop scheduling problem. Our problem optimizes a steel manufacturing process. A comparison on 5 qubits shows that the recent filtering variational quantum eigensolver (F-VQE) converges faster and samples the global optimum more frequently than the quantum approximate optimization algorithm (QAOA), the standard variational quantum eigensolver (VQE), and variational quantum imaginary time evolution (VarQITE). Furthermore, F-VQE readily solves problem sizes of up to 23 qubits on hardware without error mitigation post processing.
Abstract Quantum Annealing (QA) is a metaheuristic for solving optimization problems in a time-efficient manner. Therefore, quantum mechanical effects are used to compute and evaluate many possible solutions of an optimization problem simultaneously. Recent studies have shown the potential of QA for solving such complex assignment problems within milliseconds. This also applies for the field of job shop scheduling, where the existing approaches however focus on small problem sizes. To assess the full potential of QA in this area for industry-scale problem formulations, it is necessary to consider larger problem instances and to evaluate the potentials of computing these job shop scheduling problems while finding a near-optimal solution in a time-efficient manner. Consequently, this paper presents a QA-based job shop scheduling. In particular, flexible job shop scheduling problems in various sizes are computed with QA, demonstrating the efficiency of the approach regarding scalability, solutions quality, and computing time. For the evaluation of the proposed approach, the solutions are compared in a scientific benchmark with state-of-the-art algorithms for solving flexible job shop scheduling problems. The results indicate that QA has the potential for solving flexible job shop scheduling problems in a time efficient manner. Even large problem instances can be computed within seconds, which offers the possibility for application in industry.
The Job Shop Scheduling Problem (JSSP) has always been considered as one of the most complex and industry essential scheduling problems. Optimizing the makespan of a given schedule generally involves using dedicated algorithms, local search strategies, or metaheuristics. These approaches, however, heavily rely on classical computational power, which is bounded by the physical limits of microcontrollers and power issues. Inspired by the promising results achieved for Quantum Annealing (QA) based approaches to solve JSSP instances, we propose a new approach that uses gate-model quantum architecture as an alternative to QA. We find that we can make use of the time-indexed JSSP instance representation to build a cost Hamiltonian, which can be embedded into Quantum Approximate Optimization Algorithm (QAOA) to find an optimal solution to a basic JSSP instance. We demonstrate the use of QAOA to solve the JSSP, and we evaluate its efficiency and accuracy for this problem from experimental results, as there is an increased urgency to demonstrate the applicability of quantum optimization algorithms. We also find that optimal variational parameters form patterns that can facilitate computation in bigger quantum circuits. Additionally, we compare the obtained noiseless simulation results of gate-model quantum circuits demonstrating the relationship between two evaluation criteria - makespan and energy. Finally, we analyze and present the overall performance of our approach with the increasing deadline and simulated depth of QAOA circuits.
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.