# 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:
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.

## 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

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.

## 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

- Special case of [[quantum-state-graph|Graph states]] where the graph is a regular lattice
- Generalization of [[quantum-state-bell-00|Bell states]] (2-qubit cluster)
- Related to topological codes and surface codes used in quantum error correction
- Connection to [[quantum-state-ghz|GHZ]] and [[quantum-state-w|W states]] through different entanglement patterns