I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'.
"The number of repetitions depends on the desired state $\rho$. In the worst case, it is $O(d)$, but in many cases of practical interest, it is much smaller. For example, for stabilizer states, the number of repetitions is constant, independent of the size of the system..."
Let $\mathcal{G}_n$ denote the Pauli group on $n$ qubits. An $n$-qubit state $|\psi\rangle$ is called a stabilizer state if there exists a subgroup $S \subset \mathcal{G}_n$ such that $|S|=2^n$ and $A|\psi\rangle = |\psi\rangle$ for every $A\in S$.
For example, $(|00\rangle+|11\rangle)/\sqrt2$ is a stabilizer state, because it is a $+1$ eigenstate of the elements of the following four-element subgroup of $\mathcal{G}_2$: $\{II, XX, -YY, ZZ\}$.
Stabilizer states have a number of interesting properties. For example, they are exactly the states that are reachable from $|0\dots 0\rangle$ using the Clifford gates and thus, by Gottesman-Knill theorem, any quantum computation that takes place entirely in the set of stabilizer states can be simulated efficiently on a classical computer.
The significance of stabilizer states in Direct Fidelity Estimation (DFE) lies in the fact that they are a prime example of well-conditioned states. The cost of DFE on such states is relatively low.