Cluster state

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. 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 | ϕ { κ } ⟩ C {displaystyle |phi _{{kappa }} angle _{C}} are states which obey the set eigenvalue equations: where K ( a ) {displaystyle K^{(a)}} are the correlation operators with σ x {displaystyle sigma _{x}} and σ z {displaystyle sigma _{z}} being Pauli matrices, N ( a ) {displaystyle N(a)} denoting the neighbourhood of a {displaystyle a} and { κ a ∈ { 0 , 1 } | a ∈ C } {displaystyle {kappa _{a}in {0,1}|ain C}} 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 ofcold atoms. Here are some examples of one-dimensional cluster states (d=1), for n = 2 , 3 , 4 {displaystyle n=2,3,4} , where n {displaystyle n} is the number of qubits. We take κ a = 0 {displaystyle kappa _{a}=0} for all a {displaystyle a} , which means the cluster state is the unique simultaneous eigenstate that has corresponding eigenvalue 1 under all correlation operators. In each example the set of correlation operators { K ( a ) } a {displaystyle {K^{(a)}}_{a}} and the corresponding cluster state is listed. In all examples I {displaystyle I} is the identity operator, and tensor products are omitted. The states above can be obtained from the all zero state | 0 … 0 ⟩ {displaystyle |0ldots 0 angle } by first applying a Hadamard gate to every qubit, and then a controled-Z gate between all qubits that are adjacent to each other.