# Exponential-time quantum algorithms for graph coloring problems

## TL;DR

This paper introduces new exponential-time quantum algorithms for graph coloring problems, significantly improving upon classical algorithms by leveraging quantum memory and search techniques.

## Contribution

It presents the first quantum algorithms for graph coloring that outperform classical methods in exponential time complexity.

## Key findings

- Quantum algorithm for chromatic number with $O(1.9140^n)$ time
- Quantum algorithm for 20-coloring with $O(1.9575^n)$ time
- Quantum techniques improve classical exponential algorithms quadratically

## 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 $O(1.9575^n)$. In the polynomial-space quantum algorithm, we essentially show $(4-\epsilon)^n$-time classical algorithms that can be improved quadratically by Grover's search.