In this paper, we initiate the study of quantum algorithms in the Graph Drawing research area. We focus on two foundational drawing standards: 2-level drawings and book layouts. Concerning $2$-level drawings, we consider the problems of obtaining drawings with the minimum number of crossings, $k$-planar drawings, quasi-planar drawings, and the problem of removing the minimum number of edges to obtain a $2$-level planar graph. Concerning book layouts, we consider the problems of obtaining $1$-page book layouts with the minimum number of crossings, book embeddings with the minimum number of pages, and the problem of removing the minimum number of edges to obtain an outerplanar graph. We explore both the quantum circuit and the quantum annealing models of computation. In the quantum circuit model, we provide an algorithmic framework based on Grover's quantum search, which allows us to obtain, at least, a quadratic speedup on the best classical exact algorithms for all the considered problems. In the quantum annealing model, we perform experiments on the quantum processing unit provided by D-Wave, focusing on the classical $2$-level crossing minimization problem, demonstrating that quantum annealing is competitive with respect to classical algorithms.

翻译：本文我们首次在图形绘制研究领域中开展量子算法的研究。我们聚焦于两种基础绘制标准：双层绘制和书式布局。关于双层绘制，我们考虑了以下问题：获取交叉点数量最少的绘制、k-平面绘制、准平面绘制，以及通过移除最少边数来获得双层平面图的问题。关于书式布局，我们考虑以下问题：获取交叉点数量最少的单页书式布局、页数最少的书式嵌入，以及通过移除最少边数来获得外平面图的问题。我们同时探索了量子电路和量子退火两种计算模型。在量子电路模型中，我们基于Grover量子搜索提出了一种算法框架，该框架使得所有考虑问题的经典精确算法至少能获得二次加速。在量子退火模型中，我们在D-Wave提供的量子处理单元上进行实验，重点研究经典的二层交叉最小化问题，实验证明量子退火与经典算法相比具有竞争力。