Computational Time Complexity for k-Sum Problem Amalgamated with Quantum Search

Abstract

The k - Sum Problem, which is a generic member of the family of which 2 - Sum and 3 - Sum problems are the youngest siblings is one of the most interesting problems in the domain of Optimization Techniques. Many researchers have shown that the k - Sum problem can be solved in no less than the order of $n_{k-1}$. On the other side, many researchers have tried and have successfully minimized its Computational Complexity, though quite negligibly. But since any subtle method doesn’t exist to minimize its Computational Complexity by a major pie, the Query - “Can k - Sum problem be solved in $O(n_{k-1-\epsilon})$ for some $\epsilon \gt 0$ ” have been added in the list of UPCS (Unsolved Problems in Computer Science). In this article, we will effort to analyse the Complexity of Computing the $k - Sum$ problem, by exemplifying minimal bounds of Quantum Search, $\Omega\left(\frac{\sqrt[2]{\log _2 n}}{\log _2\left(\log _2 n\right)}\right)$ as stated by Buhrman. Now, one assumption that this minimal bound holds is that the element to be searched will be composed in some ordered manner. To extrude that, we will extend our work by making use of Grover’s Search, with Computational Complexity of the order, $O(\sqrt[2]{n})$, which is not known to make use of any prerequisite.