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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 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.
^H. J. Briegel; R. Raussendorf (2001). "Persistent Entanglement in arrays of Interacting Particles". Physical Review Letters. 86 (5): 910–3. arXiv:quant-ph/0004051. Bibcode:2001PhRvL..86..910B. doi:10.1103/PhysRevLett.86.910. PMID 11177971. S2CID 21762622.
^Briegel, Hans J. (12 August 2009). "Cluster States". In Greenberger, Daniel; Hentschel, Klaus & Weinert, Friedel (eds.). Compendium of Quantum Physics - Concepts, Experiments, History and Philosophy. Springer. pp. 96–105. ISBN 978-3-540-70622-9.
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