Chapter 12 - Cluster State-based Quantum Computing

Abstract

This chapter is devoted to cluster state-based quantum computing. The chapter starts with a definition of cluster states, followed by a description of their relationship to graph states and stabilizer formalism. We also discuss the relationship between cluster states and qubit teleportation circuits. We further formulate the conditions to be met for a corresponding set of unitaries to be able to generate the cluster state. We then relate the problem of generating the cluster state to the Ising model. The focus is then moved to the universality of cluster state-based quantum computing. We prove that a 1D cluster state is sufficient to implement an arbitrary single-qubit gate, while a 2D cluster state is needed for an arbitrary two-qubit gate by applying the proper sequence of measurements. We demonstrate that a five-qubit linear chain cluster is enough to implement arbitrary single-qubit rotation by employing the Euler decomposition theorem. We also demonstrate that T-shape and H-shape cluster states can be used to implement the CNOT gate. We then discuss cluster state processing by first describing the roles of X-, Z-, and Y-measurements. We further provide a generic 2D cluster state to be used in one-way quantum computation (1WQC) and provide details of the corresponding quantum computational model. During one-way quantum computation, the random Pauli by-product operator naturally arises, and we show that the performance of 1WQC is not affected by it. After that we discuss various physical implementations suitable for 1WQC with special focus being devoted to photonic 1WQC implementations, including resource-efficient linear optics implementation. We describe how the Bell states can be used to form linear chain, T-shape, H-shape, and arbitrary 2D cluster states through type I and type II fusion processes. We describe how the implementation of cluster states and 1WQC can be experimentally demonstrated. Furthermore, we briefly describe the basic concepts of continuous variable cluster state-based quantum computing. Finally, after the chapter summary, we provide a set of problems for self-study.