# What is a stabilizer state?

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.

• great, thanks! Very well explained. If possible, can you also provide a reference to the definition? as in, a link to a textbook or paper? Jul 1, 2021 at 18:05
• The concept is part of the stabilizer formalism for which a good reference is Daniel Gottesman's PhD thesis (even though its focus is on error correction and consequently on 2- and more dimensional stabilizer subspaces rather than 1-dimensional subspaces, i.e. stabilizer states). Also, as is the case with many fundamental concepts in QC, stabilizer formalism is introduced in Nielsen & Chuang. See section 10.5.1 on p.454. Finally, here is an example of a paper focused on stabilizer states themselves. Jul 1, 2021 at 18:44
• great thanks!!! I read most of the section in Neilson and Chuang, prior to asking the question, but there was no mention stabilizer of states anywhere in the text book. Maybe I missed it?... Jul 2, 2021 at 22:05
• I think you're right: the index in N&C does not list the term, though they do define terms such as "stabilizer code". The original book is now over two decades old (1st ed edition is from 2000, 2nd edition is from 2010), so perhaps the term "stabilizer state" has emerged following the publication. Wrote an answer to your other question. Jul 2, 2021 at 22:57
• In your answer you describe $\{ II,XX,YY,ZZ \}$ as a four element subgroup, so it seems that you are using the convention $Y=XZ$. However in your comment about single qubit stabilizer states you claim that $\pm Y$ has a $+1$ eigenstate, which would imply you are using the convention $Y=iXZ$. Maybe I'm missing something but I think there is a small inconsistency here? Aug 19, 2022 at 15:51