# Does the $|W\rangle$ state have a stabilizer?

I've been practicing trying to find stabilizer groups for various multi-qubit states $|\psi\rangle$.

So far the only method I know is to start with the state $|0\rangle^n$ which is stabilized by the generating set $\lbrace Z_k\rbrace_{k=1}^n$, (where $Z_k$ is the pauli-Z acting on the kth qubit), and then finding a circuit that constructs $|\psi\rangle$, and evolving the stabilizer generators $S_i \to US_i U^{-1}$ for each gate $U$ in the circuit.

However, as the answer to this post hints, the $W$-state

\begin{align} |W_3\rangle = \frac{|100\rangle + |010\rangle + |001\rangle}{\sqrt{3}} \end{align}

requires non-Clifford gates in its construction, so it seems like you would end up with stabilizers that aren't a subgroup of the Pauli group (even though by definition Stabilizers must be subgroups of the Pauli group).

So I guess my questions are

• Do there exist stabilizers for $|W_3\rangle$ (or in general, any arbitrary state?)

• If not, how can you tell if a given state can or cannot be stabilized?

• Given an arbitrary state $|\psi\rangle$, is the best way to find the generating stabilizers by following through a circuit that constructs $|\psi\rangle$, or is there a more straightforward way?