DNA computation (DNAC) has been proposed to solve the satisfiability (SAT) problem due to operations in parallel on extremely large numbers of strands. This paper attempts to treat the DNA-based bio-molecular solution for the SAT problem from the quantum mechanical perspective with a purpose to explore the relationship between DNAC and quantum computation (QC). To achieve this goal, it first builds up the correspondence of operations between QC and DNAC. Then it gives an example for the case of two variables and three clauses for details of this theory. It also demonstrates a three-qubit experiment for solving the simplest SAT problem with a single variable on a liquid-state nuclear magnetic resonance ensemble to verify this theory. Some discussions are made for the potential application and for further exploration of the present work.
DNA computation could in principle solve the satisfiability (SAT) problem due to the operations in parallel on extremely large numbers of strands. We demonstrate some quantum gates corresponding to the DNA ones, based on which an implementation of DNA algorithm for SAT problem is available by quantum mechanical way. Since quantum computation owns the favorable feature of operations in parallel on 2
n states by using only n qubits, instead of 2
n strands in DNA computation, computational complexity is much reduced in treating the SAT problem quantum mechanically. We take a three-clause SAT problem with two variables as an example, and carry out a NMR experiment for solving a one-variable SAT problem.
This paper aims to propose solution for a classical and difficult decision problem using quantum computing. At first, it provides experimental samples taking well-known search problems, where it is possible to establish a plausible comparison between classical search algorithms and a similar quantum algorithm. Moreover, a quantum algorithm is construct in order to deliver solution for a sophisticated mathematical problem called N-Queens. The algorithm was implemented in a quantum computing simulator obtaining solutions for problems with up to 128 queens at a reasonable cost.
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.
In this paper, we propose a bio-molecular algorithm with O(${n}$2) 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
2nitems of
nbits. We show that the value of each bit of the outcome is determined by executing our bio-molecular algorithm
ntimes. Then, we show how to view a bio-molecular solution space with
2n-1DNA 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-1DNA strands. Next, we demonstrate that the value of each bit of the output solution can be determined by executing the proposed extended quantum algorithms
ntimes. To verify our theorem, we find the maximum-sized clique to a graph with two vertices and one edge and the solution
bthat satisfies
b2≡1(mod 15) and
1<b<(15/2)using IBM Quantum’s backend.
Quantum computers can potentially solve problems that are computationally intractable on a classical computer in polynomial time using quantum-mechanical effects such as superposition and entanglement. The N-Queens Problem is a notable example that falls under the class of NP-complete problems. It involves the arrangement of N chess queens on an N x N chessboard such that no queen attacks any other queen, i.e. no two queens are placed along the same row, column or diagonal. The best time complexity that a classical computer has achieved so far in generating all solutions of the N-Queens Problem is of the order O(N!). In this paper, we propose a new algorithm to generate all solutions to the N-Queens Problem for a given N in polynomial time of order O(N^3) and polynomial memory of order O(N^2) on a quantum computer. We simulate the 4-queens problem and demonstrate its application to satellite communication using IBM Quantum Experience platform.