Exponential-Time Quantum Algorithms for Graph Coloring Problems

Abstract

The fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time \(\Omega (2^n)\) for \(k\ge 5\). In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time \(O(1.9140^n)\) using quantum random access memory (QRAM). Our approach is based on Ambainis et al.’s quantum dynamic programming with applications of Grover’s search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 20-coloring problem with running time \(O(1.9575^n)\). For the polynomial-space quantum algorithm, we essentially show \((4-\epsilon )^n\)-time classical algorithms that can be improved quadratically by Grover’s search.