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 possessing a large number of accessible internal degrees of freedom that are controlled with sufficient accuracy. This work leverages the principle of superposition to solve the optimization problem. 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 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 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 is benchmarked by a well-known classical algorithm (branch and bound) in terms of the number of steps needed for convergence to the solution. This approach carries the potential to improve the solutions obtained for larger-size problems using hybrid quantum–classical algorithms.

中文翻译：

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使用单个原子的整数规划


整数规划（IP），顾名思义，是一种基于整数变量的方法，通常用于制定具有约束的现实世界优化问题。目前，量子算法通过使用二进制变量将IP重新表示为无约束的形式，这是一种间接且消耗资源的解决方法。我们开发了一种算法，可以将原始形式的 IP 问题映射并解决到任何拥有大量可访问的内部自由度且受足够精度控制的量子系统。这项工作利用叠加原理来解决优化问题。以单个里德伯原子为例，我们将整数值与属于不同流形的电子态相关联，并实现不同态的选择性叠加来解决完整的IP问题。对于具有最多八个变量和四个约束的典型 IP 问题，可在几微秒内找到最佳解决方案。这还包括非线性 IP 问题，与线性算法相比，这些问题通常更难用经典算法解决。我们的 IP 求解算法在收敛到解决方案所需的步骤数方面以著名的经典算法（分支定界）为基准。这种方法有可能改进使用混合量子经典算法获得的较大规模问题的解决方案。