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 that 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 a few microseconds for prototypical IP problems with up to eight variables and a maximum number of four constraints. This also includes non-linear IP problems, which are usually harder to solve with classical algorithms when compared to their linear counterparts. Our algorithm for solving IP outperforms a well-known classical algorithm (branch and bound) in terms of the number of steps needed for convergence to the solution. Our approach carries the potential to improve bounds on the solution for larger problems when compared to the classical algorithms.

翻译：整数规划（IP），顾名思义是一种基于整数变量的方法，常用于构建具有约束条件的现实优化问题。目前，量子算法通过使用二进制变量将整数规划问题重构为无约束形式，这是一种间接且消耗资源的求解方式。我们开发了一种算法，可将整数规划问题以其原始形式映射到任何拥有大量可访问且能被精准控制的内自由度的量子系统中。以单个里德伯原子为例，我们将整数值与属于不同能级库的电子态相关联，并实现这些不同态的选择性叠加，从而解决完整的整数规划问题。对于包含最多八个变量和四个约束的原型整数规划问题，我们能在几微秒内找到最优解。这还包括非线性整数规划问题——相比线性问题，经典算法通常更难求解此类问题。在收敛到解所需的步数方面，我们的整数规划求解算法优于知名的经典算法（分支定界法）。与经典算法相比，我们的方法在解决更大规模问题时具有提升解边界的潜力。