Quantum Speedup and Mathematical Solutions of Implementing Bio-Molecular Solutions for the Independent Set Problem on IBM Quantum Computers
Weng-Long Chang, Ju-Chin Chen, Wen‐Yu Chung, Chun-Yuan Hsiao, Renata Wong, Athanasios V. Vasilakos · IEEE Transactions on NanoBioscience · 2021
In this paper, we propose a bio-molecular algorithm with O( n2+ m) biological operations, O( 2n) DNA strands, O( n) tubes and the longest DNA strand, O( n), for solving the independent-set problem for any graph G with m edges and n vertices. Next, we show that a new kind of the straightforward Boolean circuit yielded from the bio-molecular solutions with m NAND gates, ( m +n × ( n + 1)) AND gates and (( n × ( n + 1))/2) NOT gates can find the maximal independent-set(s) to the independent-set problem for any graph G with m edges and n vertices. We show that a new kind of the proposed quantum-molecular algorithm can find the maximal independent set(s) with the lower bound Ω( 2n/2) queries and the upper bound O( 2n/2) queries. This work offers an obvious evidence for that to solve the independent-set problem in any graph G with m edges and n vertices, bio-molecular computers are able to generate a new kind of the straightforward Boolean circuit such that by means of implementing it quantum computers can give a quadratic speed-up. This work also offers one obvious evidence that quantum computers can significantly accelerate the speed and enhance the scalability of bio-molecular computers. Next, the element distinctness problem with input of n bits is to determine whether the given 2nreal numbers are distinct or not. The quantum lower bound of solving the element distinctness problem is Ω( 2n×(2/3)) queries in the case of a quantum walk algorithm. We further show that the proposed quantum-molecular algorithm reduces the quantum lower bound to Ω(( 2n/2)/( 21/2)) queries. Furthermore, to justify the feasibility of the proposed quantum-molecular algorithm, we successfully solve a typical independent set problem for a graph G with two vertices and one edge by carrying out experiments on the backend ibmqx4 with five quantum bits and the backend simulator with 32 quantum bits on IBM's quantum computer.