# Fake Stabilizer States

Consider an $$n$$ qubit stabilizer state $$|\psi>$$ with stabilizer group $$S$$. Then the projector onto this one dimensional stabilizer subspace is given by $$\sum_{g \in S} g$$ $$|\psi>$$ can always be obtained by applying the projector to some computational basis ket $$|b>$$ ($$b$$ is some length $$n$$ bit string). $$|\psi>= \sum_{g \in S} g |b>$$ Every $$g \in S$$ can be written $$g=\alpha_g g_Xg_Z$$ where $$g_X$$ is a tensor products of $$X$$s and $$I$$s and $$g_Z$$ is a tensor product of $$Z$$s and $$I$$s and $$\alpha_g$$ has one of the four values $$\alpha_g= \pm 1, \pm i$$ $$\sum_{g \in S} g | b> =\sum_{g \in S} \alpha_g g_X g_Z | b >=\sum_{g \in S} (-1)^{g_Z\cdot b} \alpha_g g_X | b>= \sum_{g \in S} (-1)^{g_Z\cdot b} \alpha_g | g_X +b>$$ where $$|g_X>$$ is the computational basis ket for the bit string corresponding to $$g_X$$ (each $$I$$ is a $$0$$ each $$X$$ is a $$1$$). The $$g_X$$ that show up this way are exactly the span of the $$g_X$$ for the stabilizer generators. So the bit strings in the $$|g_X>$$ will form a binary vector space. And thus the $$|g_X+b>$$ will be exactly an affine translate of a binary vector space by the fixed vector $$b$$ (In particular, the size of the support must be $$2^{dim(C)}$$ since that is the size of a binary vector space $$C$$, so this is a proof the the fact of part ii of Thm 9 Corollary 2 discussed in Stabilizer codes and 1,-1 coefficients that the support is always a power of $$2$$ )

From this we can conclude that every stabilizer state is a superposition over the $$2^{dim(C)}$$ computational basis kets of some affine binary vector space $$C$$ with all coefficients taking one of the values $$\pm1, \pm i$$.

Indeed this question What are nontrivial examples of stabilizer codes whose codewords have some $\pm i$ coefficients? says that every stabilizer state is equivalent (by Pauli operators) to a stabilizer state with just $$\pm 1$$ coefficients (no $$\pm i$$ coefficients)

This leads me to ask: If $$|\psi>$$ is a uniform superposition over a an affine binary vector space $$C$$ with all coefficients $$\pm 1$$ then must it be the case that $$| \psi>$$ is a stabilizer state?

Call a state which is of that form but is not a stabilizer state a "fake stabilizer state"

Question: Do fake stabilizer states exist?

Note: Every single qubit fake stabilizer state is an actual stabilizer state. Up to global scalar the six possible single qubit fake stabilizer states (including the ones with $$i$$ here for the sake of completeness) are $$|0>\;,\;|1>\;,\;|0>+|1>\;,\;|0>-|1>\;,\;|0>+i|1>\;,\;|0>-i|1>$$ These are stabilizer states for
$$Z,-Z,X,-X,Y,-Y$$ respectively (where here we use standard convention $$Y=iXZ$$)

Edit: I looked into this a bit more and it seems that an $$n$$ qubit state is a stabilizer state if and only if it is of the form $$\frac{1}{\sqrt{2^k}}\sum_{u \in \mathbb{F}_2^k} i^{\ell(y)} (-1)^{q(y)} |y=Ru+t \rangle$$ for some vector $$t \in \mathbb{F}_2^n$$, $$n \times k$$ binary matrix $$R$$, linear function $$\ell$$ and quadratic function $$q$$. This result is originally theorem 5 of

https://arxiv.org/pdf/quant-ph/0304125.pdf

but is given in a slightly more digestible form in the appendix of

https://arxiv.org/pdf/0811.0898.pdf

So the essence of the counterexample given in the answer is that the distribution of $$+1,-1$$ coefficients does not correspond to any quadratic function $$\mathbb{F}_2^3 \to \mathbb{F}_2$$.

• I have three clarifying questions: 1.) I'm not sure I follow how you removed $g_Z$, as for example $IIZI |0010\rangle=-|0010\rangle$, so it would be more something like: $\sum_{g \in S} a_g g_X g_Z |b\rangle = \sum_{g \in S} a_g (-1)^{g_Z \cdot b} |b + g_X \rangle$. 2.) Can you motivate the "fakeness" of these states? 3.) you use k and n as well, but haven't defined k, what is it? Aug 19, 2022 at 18:16
• @BalintPato 1) good catch! 2) Ok I changed a little bit about how I describe fake states, let me know if this is more motivating 3) fixed Aug 19, 2022 at 19:09

## 1 Answer

The answer is yes, they do exist.

This answer is based on the assumed validity of https://arxiv.org/abs/1711.07848 that has an exhaustive list of 3 qubit stabilizer states in the appendix.

Here are two 3 qubit examples not contained in that table: 1--1-111 and 111-1--1 (using the shorthand notation in the paper, 1 means +1 coefficient for the given basis state, - means -1). Expressed as a sum:

$$\frac{1}{2 \sqrt{2}}( \sum_{x\notin S} |x\rangle - \sum_{x\in S} |x\rangle)$$ , where choices for $$S$$ corresponding to the two states above are $$\{x \in \mathbb{F}_2^3 | wt(x) = 1\}$$ and $$\{x \in \mathbb{F}_2^3 | wt(x) = 2\}$$.

Credit goes to my colleague, Andrew Nemec, who should be on SE but didn't feel like creating an account yet.