Exponential-Time Quantum Algorithms for Graph Coloring Problems

The fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time Omega(2(n)) for k >= 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 develop (4 - epsilon)(n)-time classical algorithms that can be improved quadratically by Grover's search.