Integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into an unconstrained form through the use of binary variables, which is an indirect and resource-consuming way of solving it. We develop an algorithm that maps and solves an IP problem in its original form to any quantum system that possesses a large number of accessible internal degrees of freedom which can be controlled with sufficient accuracy. Using a single Rydberg atom as an example, we associate the integer values to electronic states belonging to different manifolds and implement a selective superposition of these different states to solve the full IP problem. The optimal solution is found within 2-40{\mu}s for a few prototypical IP problems with up to eight variables and up to four constraints including a non-linear IP problem, which is usually harder to solve with classical algorithms when compared with linear IP problems. Our algorithm for solving IP is benchmarked using the Branch & Bound approach and it outperforms the classical algorithm in terms of the number of steps needed to converge and carries the potential to improve the bounds provided by the classical algorithm for larger problems.

翻译：整数规划（IP）作为一种基于整数变量的方法，常用于对含约束的实际优化问题建模。目前，量子算法通过引入二进制变量将IP问题转化为无约束形式，这种方式间接且资源消耗大。我们提出一种算法，可将IP问题以原始形式映射至任何拥有大量可控且精度足够高的内部自由度的量子系统。以单个里德伯原子为例，我们将整数值与不同能级群的电子态相关联，并通过选择性叠加这些不同态来求解完整IP问题。针对若干典型IP问题（最多包含八个变量及四个约束，包括通常比线性IP更难用经典算法求解的非线性IP问题），该算法可在2-40微秒内找到最优解。我们采用分支定界法对求解IP的算法进行基准测试，结果表明该算法在收敛所需步数上优于经典算法，并具有为更大规模问题改进经典算法所提供边界的潜力。