Algorithmic Theory of Qubit Routing

Abstract: 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.