Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 324 / 4 / b

A graph state has computational-basis expansionFor the path $G_A$ with edges $(1,2)$ and $(2,3)$,For the triangle $G_B$, the extra edge $(3,1)$ changes the phase whenever $x_1=x_3=1$, giving

The graph-state stabilizer generators of the path areand those of the triangle areAny pair of distinct generators has Pauli factors $X$ and $Z$ in exactly two common positions. Each such position contributes one minus sign on exchange, so the two signs cancel. ThusThis is the general commutativity mechanism for graph-state stabilizers associated with an undirected graph.

If $S|\alpha ⟩=|\alpha ⟩$ and $|\beta ⟩=U|\alpha ⟩$, thenThus conjugating the state conjugates its entire stabilizer group.

The triangle $G_B$ is obtained from the path $G_A$ by local complementation of a graph state at vertex $2$, which toggles the edge between its neighbors $1$ and $3$. The corresponding Local Clifford operation isDirect conjugation givesThese three commuting operators generate exactly the same stabilizer group as $S_1^B,S_2^B,S_3^B$. Thereforefor an irrelevant global phase $e^{i\varphi }$.