Quantum Algorithms and Mathematical Formulations of Biomolecular Solutions of the Vertex Cover Problem in the Finite-Dimensional Hilbert Space

In this paper, it is shown that the proposed quantum algorithm for implementing Boolean circuits generated from the DNA-based algorithm solving the vertex-cover problem of any graph ${\mbi G}$ with ${\mbi m}$ edges and ${\mbi n}$ vertices is the optimal quantum algorithm. Next, it is also demonstrated that mathematical solutions of the same biomolecular solutions are represented in terms of a unit vector in the finite-dimensional Hilbert space. Furthermore, for testing our theory, a nuclear magnetic resonance (NMR) experiment of three quantum bits to solve the simplest vertex-cover problem is completed.