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arXiv · 1907.00529

Exponential-time quantum algorithms for graph coloring problems

Kazuya Shimizu·Ryuhei Mori
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Abstract

The fastest known classical algorithm deciding the $k$-colorability of $n$-vertex graph requires running time $Ω(2^n)$ 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 $(4-ε)^n$-time classical algorithms that can be improved quadratically by Grover's search.

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Kazuya Shimizu, Ryuhei Mori. 2019-07-01. Exponential-time quantum algorithms for graph coloring problems. https://arxiv.org/abs/1907.00529

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