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Topological order in 1D Cluster state protected by symmetry · Around

2011

Topological order in 1D Cluster state protected by symmetry

Son, Wonmin, Amico, Luigi, Vedral, Vlatko

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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.

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It is worth noting that, for given ( A ¯ 1 l o c , B ¯ 1 l o c ) (\bar{A}_{1}^{loc},\bar{B}_{1}^{loc}) such a choice is not unique. However the other possible legitimate choice ( τ 1 y τ L z , τ 1 y τ L x ) (\tau_{1}^{y}\tau_{L}^{z},\tau_{1}^{y}\tau_{L}^{x}) , ( τ L y , τ 1 y τ L x ) (\tau_{L}^{y},\tau_{1}^{y}\tau_{L}^{x}) , τ L y \tau_{L}^{y} , τ 1 y τ L z \tau_{1}^{y}\tau_{L}^{z} and τ 1 y τ L x \tau_{1}^{y}\tau_{L}^{x} at a local level would lead to global symmetries that cannot protect the gs by local perturbations

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