Bioinspired Quantum Oracle Circuits for Biomolecular Solutions of the Maximum Cut Problem

Given an undirected, unweighted graph with n vertices and m edges, the maximum cut problem is to find a partition of the n vertices into disjoint subsets
${V}_{{1}}$
and
${V}_{{2}}$
such that the number of edges between them is as large as possible. Classically, it is an NP-complete problem, which has potential applications ranging from circuit layout design, statistical physics, computer vision, machine learning and network science to clustering. In this paper, we propose a biomolecular and a quantum algorithm to solve the maximum cut problem for any graph G. The quantum algorithm is inspired by the biomolecular algorithm and has a quadratic speedup over its classical counterparts, where the temporal and spatial complexities are reduced to, respectively,
${O}\text {(}\sqrt {{2}^{n}/{r}}\text {)}$
and
${O}\text {(}{m}^{{2}}\text {)}$
. With respect to oracle-related quantum algorithms for NP-complete problems, we identify our algorithm as optimal. Furthermore, to justify the feasibility of the proposed algorithm, we successfully solve a typical maximum cut problem for a graph with three vertices and two edges by carrying out experiments on IBM’s quantum simulator.