Solving Scheduling Problems with Quantum Computing: a Study on Flexible Open Shop

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.