Graph state on a four-node ring

For a graph $G=(V,E)$, the graph state is

$|G\rangle = \Big(\prod_{(i,j)\in E} \mathrm{CZ}_{ij}\Big)|+\rangle^{\otimes |V|}.$

For the 4-cycle $C_4$ with edges $\{01,12,23,30\}$, every vertex has exactly two neighbors, and the stabilizer generators are

$K_0 = X_0 Z_1 Z_3,\quad K_1 = X_1 Z_0 Z_2,\quad K_2 = X_2 Z_1 Z_3,\quad K_3 = X_3 Z_0 Z_2.$

Contrast with the open chain. The path-graph cluster state's end qubits have only one neighbor (generators $X_1Z_2$ and $Z_{n-1}X_n$ have weight 2), while every ring generator has weight 3 — the extra wrap-around edge removes the boundary effect and makes the stabilizer group translationally invariant under cyclic qubit relabeling.

Verification structure. Because $G$ is bipartite and CZ is diagonal, $K_i$'s commute pairwise: two generators either act on disjoint qubits, or share exactly the pair of sites where one contributes $X$ and the other $Z$ on the same site, and $XZ = -ZX$ on that one shared site is compensated by the same relation on the other shared site, giving an overall commutator of $(-1)^2=1$. This is the general argument for why any graph state's generator set $\{K_v = X_v\prod_{u\in N(v)}Z_u\}$ is a valid stabilizer group — it does not depend on the specific graph topology, only on CZ being diagonal and self-inverse.

Significance. Ring graph states are the standard test case for verifying entanglement-witness and stabilizer-measurement protocols on ring-connectivity hardware (many superconducting and trapped-ion layouts are natively rings or short chains), and they are the resource state for small quantum repeater and secret-sharing demonstrations.