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1) This equation is at best confusing, and at worst perhaps even straight-up wrong. Firstly, they use a different edge contraction notation as in the equations description and secondly, as you observe, they appear to have a Z operator acting upon qubit $v^{\prime\prime}$ which is no longer a vertex of the edge-contracted state. Perhaps the Z-operators on ...

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If you start with one graph state, which is an eigenstate of stabilizer (each of which comprises an X tensored with a bunch of Zs), then the other eigenstates of those stabilizers are the original state acted on by Z rotations (take all possible combinations). To see this note the commutation and anticommutation relations between a tensor product of Zs and ...

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I'm not sure I understand the question, since this seems quite straightforward. Graph states are Clifford states, so for a state on $n$ qubits, the set of stabilizers has $n$ generators looking like $$i^k P_1\otimes\ldots\otimes P_n$$ where $k\in\{0,1,2,3\}$ and the $P$'s are Pauli $X,Y,$ or $Z$ operators. (For graph states, these stabilizers look like \$\...

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