Cluster state

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In quantum information and quantum computing, a cluster state^[1] is a type of highly entangled state of multiple qubits. Cluster states are generated in lattices of qubits with Ising type interactions. A cluster C is a connected subset of a d-dimensional lattice, and a cluster state is a pure state of the qubits located on C. They are different from other types of entangled states such as GHZ states or W states in that it is more difficult to eliminate quantum entanglement (via projective measurements) in the case of cluster states. Another way of thinking of cluster states is as a particular instance of graph states, where the underlying graph is a connected subset of a d-dimensional lattice. Cluster states are especially useful in the context of the one-way quantum computer. For a comprehensible introduction to the topic see.^[2]

Formally, cluster states ${\displaystyle |\phi _{\{\kappa \}}\rangle _{C}}$ are states which obey the set eigenvalue equations:

${\displaystyle K^{(a)}{\left|\phi _{\{\kappa \}}\right\rangle _{C}}=(-1)^{\kappa _{a}}{\left|\phi _{\{\kappa \}}\right\rangle _{C}}}$

where ${\displaystyle K^{(a)}}$ are the correlation operators

${\displaystyle K^{(a)}=\sigma _{x}^{(a)}\bigotimes _{b\in \mathrm {N} (a)}\sigma _{z}^{(b)}}$

with ${\displaystyle \sigma _{x}}$ and ${\displaystyle \sigma _{z}}$ being Pauli matrices, ${\displaystyle N(a)}$ denoting the neighbourhood of ${\displaystyle a}$ and ${\displaystyle \{\kappa _{a}\in \{0,1\}|a\in C\}}$ being a set of binary parameters specifying the particular instance of a cluster state.