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IEEE Transactions on NanoBioscience

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

Weng-Long Chang, Wen-Yu Chung, Chun-Yuan Hsiao, Renata Wong, Ju-Chin Chen, Mang Feng, Athanasios V. Vasilakos

Published Nov 25, 2021• 14 citations
Publisher

Abstract

In this paper, we propose a bio-molecular algorithm with O(${n}$2) biological operations, O($2^{n-1}$) 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$2^{n}$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$2^{{\textit {n-1}}}$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$2^{{\textit {n-1}}}$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${b}^{2} \equiv 1$(mod 15) and$1 < {b} < (15/2)$using IBM Quantum’s backend.

DOI
10.1109/tnb.2021.3130811

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We demonstrate that the logic computation performed by the DNA-based algorithm for solving general cases of the satisfiability problem can be implemented by our proposed quantum algorithm on the quantum machine proposed by Deutsch. Moreover, we also prove that the logic computation by the bio-molecular operations proposed by Adleman can be implemented by quantum gates (for example, the Hadamard gate, NOT, CNOT, and CCNOT) on the quantum machine. Furthermore, those NP-complete problems solved on a bio-molecular computer are also solvable on a quantum computer. To test our theory, we carry out a three-qubit NMR experiment for solving the simplest satisfiability problem.
We consider the problem of finding one or more desired items out of an unsorted database. Patel has shown that if the database permits quantum queries, then mere digitization is sufficient for efficient search for one desired item. The algorithm, called factorized quantum search algorithm, presented by him can locate the desired item in an unsorted database using O(log4N)O(log_{4}N)O(log4​N) queries to factorized oracles. But the algorithm requires that all the property values must be distinct from each other. In this paper, we discuss how to make a database satisfy the requirements, and present a quantum search engine based on the algorithm. Our goal is achieved by introducing auxiliary files for the property values that are not distinct, and converting every complex query request into a sequence of calls to factorized quantum search algorithm. The query complexity of our algorithm is O(P∗Q∗M∗log4N)O(P*Q*M*log_{4}N)O(P∗Q∗M∗log4​N), where P is the number of the potential simple query requests in the complex query request, Q is the maximum number of calls to the factorized quantum search algorithm of the simple queries, M is the number of the auxiliary files for the property on which our algorithm are searching for desired items. This implies that to manage an unsorted database on an actual quantum computer is possible and efficient.