Exponential-Time Quantum Algorithms for Graph Coloring Problems

Abstract

AbstractThe fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time $$\varOmega (2^n)$$ Ω ( 2 n ) for $$k\ge 5$$ k ≥ 5 . In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time $$O(1.9140^n)$$ 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 $$O(1.9575^n)$$ O ( 1 . 9575 n ) . For the polynomial-space quantum algorithm, we essentially develop $$(4-\epsilon )^n$$ ( 4 - ϵ ) n -time classical algorithms that can be improved quadratically by Grover’s search.