Applying the Hubbard-Stratonovich Transformation to Solve Scheduling Problems Under Inequality Constraints With Quantum Annealing

Quantum annealing is a global optimization algorithm that uses the quantum tunneling effect to speed-up the search for an optimal solution. Its current hardware implementation relies on D-Wave’s Quantum Processing Units, which are limited in terms of number of qubits and architecture while being restricted to solving quadratic unconstrained binary optimization (QUBO) problems. Consequently, previous applications of quantum annealing to real-life use cases have focused on problems that are either native QUBO or close to native QUBO. By contrast, in this paper we propose to tackle inequality constraints and non-quadratic terms. We demonstrate how to handle them with a realistic use case-a bus charging scheduling problem. First, we reformulate the original integer programming problem into a QUBO with the penalty method and directly solve it on a D-Wave machine. In a second approach, we dualize the problem by performing the Hubbard-Stratonovich transformation. The dual problem is solved indirectly by combining quantum annealing and adaptive classical gradient-descent optimizer. Whereas the penalty method is severely limited by the connectivity of the realistic device, we show experimentally that the indirect approach is able to solve problems of a larger size, offering thus a better scaling. Hence, the implementation of the Hubbard-Stratonovich transformation carried out in this paper on a scheduling use case suggests that this approach could be investigated further and applied to a variety of real-life integer programming problems under multiple constraints to lower the cost of mapping to QUBO, a key step towards the near-term practical application of quantum computing.