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 $\Omega(2<sup>n)$ for $k\ge 5$. In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time $O(1.9140<sup>n)$ 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 $O(1.9575<sup>n)$. In the polynomial-space quantum algorithm, we essentially show $(4-\epsilon)<sup>n$-time classical algorithms that can be improved quadratically by Grover's search.