Quantum Algorithms of Bio-molecular Solutions for the Clique Problem on a Quantum Computer

In this paper, it is demonstrated that the DNA-based algorithm [Ho et al. 2005] for solving an instance of the clique problem to any a graph G = (V, E) with n vertices and p edges and its complementary graph G1 = (V, E1) with n vertices and m = (((n*(n-1))/2)-p) edges can be implemented by Hadamard gates, NOT gates, CNOT gates, CCNOT gates, Grover's operators, and quantum measurements on a quantum computer. It is also demonstrated that if Grovers algorithm is employed to accomplish the readout step in the DNA-based algorithm, the quantum implementation of the DNA-based algorithm is equivalent to the oracle work (in the language of Grover's algorithm), that is, the target state labeling preceding Grover,s searching steps. It is shown that one oracle work can be completed with O((2 * n) * (n + 1) * (n + 2) / 3) NOT gates, one CNOT gate and O((4 * m) + (((2 * n) * (n + 1) * (n + 14)) / 6)) CCNOT gates. This is to say that for the quantum implementation of the DNA-based algorithm [Ho et al. 2005] a faster labeling of the target state is attained, which also implies a speedy solution to an instance of the clique problem

Constructing Bio-molecular Databases on a DNA-based Computer

Codd [Codd 1970] wrote the first paper in which the model of a relational database was proposed. Adleman [Adleman 1994] wrote the first paper in which DNA strands in a test tube were used to solve an instance of the Hamiltonian path problem. From [Adleman 1994], it is obviously indicated that for storing information in molecules of DNA allows for an information density of approximately 1 bit per cubic nm (nanometer) and a dramatic improvement over existing storage media such as video tape which store information at a density of approximately 1 bit per 1012 cubic nanometers. This paper demonstrates that biological operations can be applied to construct bio-molecular databases where data records in relational tables are encoded as DNA strands. In order to achieve the goal, DNA algorithms are proposed to perform eight operations of relational algebra (calculus) on bio-molecular relational databases, which include Cartesian product, union, set difference, selection, projection, intersection, join and division. Furthermore, this work presents clear evidence of the ability of molecular computing to perform data retrieval operations on bio-molecular relational databases.Comment: The article includes 35 pages, several tables and figure