In quantum information and quantum computing, a cluster state 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.
Formally, cluster states are states which obey the set eigenvalue equations:
where are the correlation operators
with and being Pauli matrices, denoting the neighbourhood of and being a set of binary parameters specifying the particular instance of a cluster state.
Cluster states have been realized experimentally. They have been obtained in photonic experiments using parametric downconversion  . They have been created also in optical lattices of cold atoms .
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