Cluster State
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).
Definition
A cluster state on a 1D chain of $n$ qubits is created by:
Preparing all qubits in state $|+\rangle$
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.
Structure
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
Properties
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
Measurement-Based Quantum Computing
Prepare cluster state
Measure qubits sequentially in chosen bases (X, Y, or Z basis)
Measurement outcomes reveal errors and determine subsequent measurement angles
Final qubit state encodes computation result
This paradigm shifts computation from unitary gates applied to entangled state to adaptive measurements.
Advantages
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
Implementation
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
Relation to Other States