Stabilizer state is a quantum state defined as the +1 eigenstate of a set of commuting Pauli operators called stabilizers. Stabilizer formalism is the mathematical foundation for understanding quantum error correction, and stabilizer states can be efficiently simulated classically.

A state $|\psi\rangle$ is a stabilizer state if there exist commuting Pauli operators $S_1, S_2, \ldots, S_k$ (the stabilizers) such that $S_i|\psi\rangle = |\psi\rangle$ for all $i$. The stabilizer group $G = \langle S_1, \ldots, S_k \rangle$ is generated by these stabilizers.

• Computational basis state $|0\rangle$: stabilized by Z (since $Z|0\rangle = |0\rangle$)
• Plus state $|+\rangle$: stabilized by X (since $X|+\rangle = |+\rangle$)
• Bell state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$: stabilized by $Z_1 Z_2$ and $X_1 X_2$
• GHZ state: stabilized by $Z_1 Z_2$, $Z_2 Z_3$, and $X_1 X_2 X_3$

Quantum error correcting codes are built from stabilizer states. Code space is the subspace stabilized by the code's stabilizer generators. Syndrome measurement projects onto code space and reveals error information without disturbing the state.

Stabilizer states with Clifford operations can be simulated classically in polynomial time. A circuit using only Clifford gates and measuring stabilizer eigenstates offers no quantum advantage. Adding non-Clifford gates (like T) breaks simulability.

Measuring an observable $M$ whose eigenspaces are stabilizer states yields outcome +1 or -1 corresponding to eigenspaces. This is the basis of syndrome measurement in error correction.

Stabilizer formalism cannot represent generic quantum states. A random state requires exponentially many stabilizer generators to describe. States outside the stabilizer subspace require non-Clifford operations to reach.