One-dimensional cluster state

A 1D cluster state on $n$ qubits is prepared by

$|C_n\rangle = \Big(\prod_{i=1}^{n-1} \mathrm{CZ}_{i,i+1}\Big) |+\rangle^{\otimes n}, \qquad |+\rangle = \frac{|0\rangle+|1\rangle}{\sqrt2}.$

Stabilizer generators. For the path graph with vertices $1,\dots,n$, the state is the unique joint $+1$ eigenstate of the $n$ commuting generators

$K_i = X_i \prod_{j \in N(i)} Z_j, \qquad N(1)=\{2\},\; N(n)=\{n-1\},\; N(i)=\{i-1,i+1\}\ \text{otherwise}.$

For $n=4$ explicitly: $K_1 = X_1Z_2$, $K_2 = Z_1X_2Z_3$, $K_3 = Z_2X_3Z_4$, $K_4 = Z_3X_4$. Each $K_i$ squares to identity, all pairs commute (any two generators either share zero neighboring sites or share exactly two anti-commuting Pauli factors that cancel), and their product group has $2^n$ elements — the full stabilizer group of a pure $n$-qubit state.

Why it is Clifford, not universal by itself. CZ and $H$ are both Clifford operations, so the cluster state itself carries no non-Clifford resource; it is entangling structure. Universality in the one-way model comes from the choice of measurement bases (rotated single-qubit measurements), which are adaptively chosen based on earlier outcomes — the classical feed-forward, not the resource state, injects non-Clifford power.

Physical significance. Raussendorf and Briegel's one-way quantum computer showed that any circuit can be compiled into: (1) prepare a large enough cluster state, (2) measure each qubit in a computed adaptive basis, (3) read off the answer from the final unmeasured qubits and classical corrections. This underlies photonic and superconducting measurement-based architectures where entangling gates are cheap up front and computation is pushed into measurement choices.