Quantum Circuit Based on Grover Algorithm to Solve Hamiltonian Cycle Problem

We propose the concepts of the explicit oracle and the implicit oracle for realizing quantum algorithms. Then, the quantum circuit of the well-known Grover algorithm is constructed with the explicit oracle to solve the Hamiltonian cycle problem for the complete graph. The quantum circuit has a quadratic speedup over the classical unstructured search algorithm for solving the same problem. The IBM quantum computer simulator is used to run the quantum circuit to validate that it can indeed derive the Hamiltonian cycle of the complete graph.