Algorithmic Theory of Qubit Routing

TL;DR

This paper analyzes the qubit routing problem in quantum computing, proving its NP-hardness, and providing fixed-parameter and polynomial-time algorithms for specific cases, advancing theoretical understanding of quantum compiler optimization.

Contribution

It establishes the NP-hardness of qubit routing and introduces new algorithms for special cases, bridging theory and quantum compiler design.

Findings

Proved qubit routing is NP-hard.

Developed a fixed-parameter algorithm based on the number of two-qubit gates.

Designed a polynomial-time algorithm for limited qubit interactions.

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.