Quantum Speedup for Inferring the Value of Each Bit of a Solution State in Unsorted Databases Using a Bio-molecular Algorithm on IBM Quantum's Computers.

In this paper, we propose a bio-molecular algorithm with O(n2) biological operations, O(2n-1) DNA strands, O(n) tubes and the longest DNA strand, O(n), for inferring the value of a bit from the only output satisfying any given condition in an unsorted database with 2n items of n bits. We show that the value of each bit of the outcome is determined by executing our bio-molecular algorithm n times. Then, we show how to view a bio-molecular solution space with 2n-1 DNA strands as an eigenvector and how to find the corresponding unitary operator and eigenvalues for inferring the value of a bit in the output. We also show that using an extension of the quantum phase estimation and quantum counting algorithms computes its unitary operator and eigenvalues from bio-molecular solution space with 2n-1 DNA strands. Next, we demonstrate that the value of each bit of the output solution can be determined by executing the proposed extended quantum algorithms n times. To verify our theorem, we find the maximum-sized clique to a graph with two vertices and one edge and the solution b that satisfies b2 ≡ 1 (mod 15) and 1 < b < (15 / 2) using IBM Quantum's backend.