The qubit routing problem, also known as the swap minimization problem, is a (classical) combinatorial optimization problem that arises in the design of compilers of quantum programs. We study the qubit routing problem from the viewpoint of theoretical computer science, while most of the existing studies investigated the practical aspects. We concentrate on the linear nearest neighbor (LNN) architectures of quantum computers, in which the graph topology is a path. Our results are three-fold. (1) We prove that the qubit routing problem is NP-hard. (2) We give a fixed-parameter algorithm when the number of two-qubit gates is a parameter. (3) We give a polynomial-time algorithm when each qubit is involved in at most one two-qubit gate.

翻译：量子比特路由问题，也称为交换最小化问题，是量子程序编译器设计中出现的一个（经典）组合优化问题。我们从理论计算机科学的角度研究量子比特路由问题，而现有研究大多关注其实践方面。我们专注于量子计算机的线性最近邻（LNN）架构，其中图拓扑为路径。我们的结果有三方面：（1）我们证明量子比特路由问题是NP难的。（2）当双量子比特门数量作为参数时，我们给出一个固定参数算法。（3）当每个量子比特最多参与一个双量子比特门时，我们给出一个多项式时间算法。