Cluster state is a multi-qubit entangled state arranged in a one-dimensional, two-dimensional, or higher-dimensional lattice where neighboring qubits are entangled via CZ interactions. Cluster states are the resource for measurement-based quantum computing (one-way quantum computing).

A cluster state on a 1D chain of $n$ qubits is created by:

1. Preparing all qubits in state $|+\rangle$
2. Applying CZ gates between neighboring pairs: $\text{CZ}_{i,i+1}$ for each neighbor pair

The resulting state exhibits long-range entanglement and is stabilized by check operators that commute with all stabilizers.

• 1D cluster: linear chain, stabilizers are $Z_i Z_{i+1}$
• 2D cluster: square lattice, enables universal quantum computation
• Graph states: generalizes to arbitrary graph structures; cluster states are one instance

• Stabilizer state: stabilized by commuting Pauli operators (efficiently simulatable with Clifford gates alone, until measurement)
• Measurement-based computation: implement quantum algorithms by measuring qubits in specified bases; outcomes determine classically post-computed corrections

1. Prepare cluster state
2. Measure qubits sequentially in chosen bases (X, Y, or Z basis)
3. Measurement outcomes reveal errors and determine subsequent measurement angles
4. Final qubit state encodes computation result

This paradigm shifts computation from unitary gates applied to entangled state to adaptive measurements.

• One-way: no need to return quantum state to original entanglement after each gate
• Topological protection: 2D cluster states admit topological error correction
• Parallelism: measurements can be parallelized unlike gate-based circuits

Cluster states are realizable on:

• Photonic systems: post-selected cluster states
• Trapped ions: CZ gates between neighboring pairs
• Superconducting qubits: mediated two-qubit interactions
• Spin systems: Heisenberg interactions create cluster-like entanglement

• Special case of Graph states where the graph is a regular lattice
• Generalization of Bell states (2-qubit cluster)
• Related to topological codes and surface codes used in quantum error correction
• Connection to GHZ and W states through different entanglement patterns