Topological order in 1D Cluster state protected by symmetry

TL;DR

This paper constructs a Z2*Z2 symmetry protecting the ground state degeneracy in 1D cluster states with open boundaries, revealing the role of stabilizers and control phase transformations in topological order.

Contribution

It introduces a method to explicitly construct the protecting symmetry for 1D cluster states, clarifying the origin of degeneracy and stabilizer structure.

Findings

Constructed Z2*Z2 symmetry protecting degeneracy

Demonstrated stabilizers do not form complete integrals of motion

Applied control phase transformations to reveal the degeneracy structure

Abstract

We demonstrate how to construct the Z2*Z2 global symmetry which protects the ground state degeneracy of cluster states for open boundary conditions. Such a degeneracy ultimately arises because the set of stabilizers do not span a complete set of integrals of motion of the cluster state Hamiltonian for open boundary conditions. By applying control phase transformations, our construction makes the stabilizers into the Pauli operators spanning the qubit Hilbert space from which the degeneracy comes.