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.

量子退火是一种全局优化算法，它利用量子隧道效应来加速寻找最优解。其当前的硬件实现依赖于 D-Wave 的量子处理单元，这些单元在量子比特数和架构方面受到限制，同时仅限于解决二次无约束二进制优化 (QUBO) 问题。因此，以前将量子退火应用于现实生活用例的重点是原生 QUBO 或接近原生 QUBO 的问题。相比之下，在本文中，我们建议解决不等式约束和非二次项。我们演示了如何通过一个现实的用例来处理它们——公交车充电调度问题。第一的，我们使用惩罚方法将原始整数规划问题重新表述为 QUBO，并直接在 D-Wave 机器上求解。在第二种方法中，我们通过执行 Hubbard-Stratonovich 变换将问题二元化。通过结合量子退火和自适应经典梯度下降优化器间接解决了对偶问题。尽管惩罚方法受到现实设备连接性的严重限制，但我们通过实验表明，间接方法能够解决更大尺寸的问题，从而提供更好的缩放。因此，