Quantum Speedup and Mathematical Solutions of Implementing Bio-Molecular Solutions for the Independent Set Problem on IBM Quantum Computers

In this paper, we propose a bio-molecular algorithm with O( ${n}^{{2}} + {m}$ ) biological operations, O( $2^{n}$ ) 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} \times $ ( ${n} + {1}$ )) AND gates and (( ${n} \times $ ( ${n} + {1}$ ))/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 $\Omega $ ( $2^{n/{2}}$ ) queries and the upper bound O( $2^{n/{2}}$ ) 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 $2^{n}$ real numbers are distinct or not. The quantum lower bound of solving the element distinctness problem is $\Omega $ ( $2^{n\times {(}{2}/{3}{)}}$ ) queries in the case of a quantum walk algorithm. We further show that the proposed quantum-molecular algorithm reduces the quantum lower bound to $\Omega $ (( $2^{n/{2}}$ )/( $2^{\text {1/2}}{)}$ ) 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.