Optimization of Quantum Encoding for Combinatorial Problems in Rydberg Atom Array

Programmable quantum systems utilizing Rydberg atoms have recently been shown to efficiently encode combinatorial optimization problems. The relationships among the variables in these optimization problems can be transformed into a maximum weighted independent set problem within the Rydberg atom system, requiring at most a quadratic overhead in terms of the number of qubits. Although experiments with current Rydberg quantum systems have demonstrated linear speedup compared to classical algorithms, the overhead in both the number of qubits and time remains substantial. To further reduce the overhead of encoding these optimization problems in Rydberg atom systems, we propose a simplification strategy and design a new heuristic algorithm that incorporates a topological sort approach to optimize the encoding process. We conduct extensive experiments on small graphs with dozens of qubits. The numerical results demonstrate the effectiveness of our algorithm in both runtime and optimization efficiency. Specifically, our algorithm reduces the number of required encoding qubits by 12.0% compared to existing methods. Additionally, our approach reduces the time complexity to a quadratic overhead, enabling experiments on larger graphs. The results further indicate that our algorithm provides significant time advantages over existing solutions.