A Bounded-error Quantum Polynomial Time Algorithm for Two Graph Bisection Problems

TL;DR

This paper introduces a quantum algorithm that efficiently approximates solutions for the NP-hard max-bisection and min-bisection graph problems within bounded error, using novel quantum techniques.

Contribution

It presents a new BQP algorithm for bisection problems that employs iterative partial negation and measurement, achieving high success probability in polynomial time.

Findings

Runs in $O(m^2)$ for sparse graphs and $O(n^4)$ for dense graphs

Achieves arbitrarily high success probability with polynomial space

Applies to general graphs by formulating as Boolean constraint satisfaction

Abstract

The aim of the paper is to propose a bounded-error quantum polynomial time (BQP) algorithm for the max-bisection and the min-bisection problems. The max-bisection and the min-bisection problems are fundamental NP-hard problems. Given a graph with even number of vertices, the aim of the max-bisection problem is to divide the vertices into two subsets of the same size to maximize the number of edges between the two subsets, while the aim of the min-bisection problem is to minimize the number of edges between the two subsets. The proposed algorithm runs in $O(m^2)$ for a graph with $m$ edges and in the worst case runs in $O(n^4)$ for a dense graph with $n$ vertices. The proposed algorithm targets a general graph by representing both problems as Boolean constraint satisfaction problems where the set of satisfied constraints are simultaneously maximized/minimized using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of $1-\epsilon$ for small $\epsilon>0$ using a polynomial space resources.