Generalized cluster states based on finite groups

Courtney G Brell

doi:10.1088/1367-2630/17/2/023029

Abstract

We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure and finite group algebras, and applying this to the cluster states to derive their generalization. We then investigate properties of these states including their projected entangled pair state representations, global symmetries, and relationship to the Kitaev quantum double models. We also discuss possible applications of these states.

Highlights

Cluster states [1, 2] are the prototypical resource for measurement-based quantum computation (MBQC) [3]
C G Brell distinction between abelian and non-abelian topological phases is manifested in Kitaev quantum double models for abelian and non-abelian groups respectively. Abelian phases such as the toric code cannot be used for quantum computation by braiding of quasi-particles, and are not known to be able to implement universal topological quantum computation via code-deformation
We explore the properties of the generalized cluster states defined in this way, such as their global symmetries and projected entangled pair state (PEPS) representations, in analogy to the qubit case [23, 24]

Summary

Introduction

Cluster states (or graph states) [1, 2] are the prototypical resource for measurement-based quantum computation (MBQC) [3]. They have many desirable features for a resource state: for example, they are the output of a finite-depth quantum circuit, and the frustration-free ground states of a (gapped) commuting local Hamiltonian Apart from their usefulness for standard MBQC, the cluster states are related to topologically ordered systems such as the toric code [4,5,6,7,8] and the colour codes [9, 10]. Abelian phases such as the toric code cannot be used for quantum computation by braiding of quasi-particles, and are not known to be able to implement universal topological quantum computation via code-deformation (though universal quantum computation can still be achieved by using non-topological operations such as magic state distillation [19, 20]) For these reasons, we define a family of generalized cluster states based on arbitrary finite groups G, where the standard qubit cluster state corresponds to the simplest group 2.

From CSS structure to finite groups

Generalized cluster states

Circuit representation

Properties of generalized cluster states

Discussion