Solving Hamiltonian Cycle Problem with Grover’s Algorithm Using Novel Quantum Circuit Designs

The Hamiltonian cycle problem (HCP) is related to the decision of whether there exists a cycle in a graph that visits every vertex exactly once. It is a well-known NP-complete problem, and thus there currently exists no classical algorithm solving it with polynomial time complexity in the worst case. With the emergence of quantum computers, the introduction of Grover’s algorithm has suggested new possibilities for solving the HCP more efficiently. In this article, we propose three novel quantum circuit designs of quantum counters to be embedded in the oracle of Grover’s algorithm to check if the input instance satisfies the HCP constraints to solve the HCP. We executed Grover’s algorithm with the novel quantum circuit designs, through the IBM Quantum Lab service, to solve the HCP to validate the design correctness.