Open-Shop Scheduling With Hard Constraints

Abstract: Encoding hard constrained optimization problems into a variational quantum algorithm often turns out to be a challenging task. In this work, we provide a solution for the class of open-shop scheduling problems (OSSP), which we achieve by rigorously employing the symmetries of the classical problem. An established approach for encoding the hard constraints of the closely related traveling salesperson problem (TSP) into mixer Hamiltonians was recently given by Hadfield et al.'s Quantum Alternating Operator Ansatz (QAOA). For OSSP, which contains TSP as a special case, we show that desired properties of similarly constructed mixers can be directly linked to a purely classical object: the group of feasibility-preserving bit value permutations. We also outline a generic way to construct QAOA-like-mixers for these problems. We further propose a new variational quantum algorithm that incorporates the underlying group structure more naturally, and implement our new algorithm for a small OSSP instance on an IBM Q System One. Unlike the QAOA, our algorithm allows bounding the amount and the domain of parameters necessary to reach every feasible solution from above: If $J$ jobs should be distributed, we need at most $J (J - 1)^{2} / {2}$ parameters.