Finding maximum independent set based on multi-stage simulated quantum adiabatic evolution

The ubiquitous nature of the maximum independent set (MIS) problem across diverse fields, including biology and chip design, presents a significant challenge. Recently, quantum algorithms have been recognized for their increased potential in addressing combinatorial optimization problems, including the MIS problem. Several quantum systems, including optical quantum circuits and Rydberg atomic systems, have been designed specifically for the MIS problem. However, elucidating the precise nature of instances that pose challenges for quantum algorithms constitutes a formidable task. In an effort to comprehend this issue, we endeavor to delineate the conditions under which instances become hard to solve by analyzing the Hamiltonian of the k-independent set (k-IS). Furthermore, we present two multi-stage strategies grounded in the simulated quantum adiabatic evolution (SQAE) of Kerr-nonlinear parametric oscillator (KPO) system for identifying the MIS in instances deemed hard to solve. More specifically, we present the algorithm leveraging SQAE of the KPO system for approximating the MIS. Our investigation of the SQAE algorithm reveals that optimal solutions are unattainable under two distinct conditions: (1) when the number of solutions is exponentially smaller to the problem’s scale, and (2) when the counts of non-optimal independent sets (IS) with m overlaps between the optimal k-IS (k>m) is exponentially smaller relative to the problem’s scale. With these two distinctive features, we propose two multi-stage SQAE strategies tailored for hard-to-solve instances. In the first approach, the SQAE algorithm is executed multiple times to identify a set of candidate vertices for the MIS. For each vertex or combinations of vertices in this set, we extract the subgraphs induced by the vertices not adjacent to them. Subsequently, the SQAE algorithm is applied to the subgraph to obtain the (k−1)-IS. In the second multi-stage strategy, we consider each vertex vi in the given graph, generating the subgraph induced by the vertices not adjacent to vi. The SQAE algorithm is then employed to find the (k−1)-IS of the subgraph. Experimental results demonstrate that both multi-stage strategies significantly enhance the performance of the SQAE algorithm for hard-to-solve instances.