Are all stabilizer codes real? Or example stabilizer codes with $\pm i$ coefficients

Question

This is a follow-up question to

Stabilizer codes and 1,-1 coefficients

A lot of well known codes (5 qubit code, 7 qubit Steane code, 9 qubit Shor code) have logical zero and logical one which can be written as (a global scalar times) a linear combination of computational basis kets with only $ \pm 1 $ as coefficients.

The question linked above shows that any stabilizer code has codewords which can be written as (a global scalar times) a linear combination of computational basis kets where every coefficient is $ \pm 1, \pm i $.

I'm curious about these $ \pm i $ coefficients. Does anyone know any stabilizer codes which seem to use $ \pm i $ in an essential way? In other words

What is an example of a stabilizer code which has codewords with some $ \pm i $ coefficients and is not equivalent by local unitaries to a stabilizer code with just $ \pm 1 $ coefficients?

In general I'm interested in any examples of cool stabilizer codes that use $ \pm i $ relative coefficients.

Note: Corollary 2 of Thm 9 in https://arxiv.org/abs/1711.07848 says some pretty cool stuff (although part (iv) of the corollary is wrong). In particular, part (iii) of Corollary 2 says that the number of $ \pm i $ amplitudes is either 0 or it is half the number of non-zero amplitudes.

Answer 1

Every stabilizer code is equivalent by local Cliffords to a real stabilizer code.

To see why observe that the logical states of the code are exactly the elements of the code space that are simultaneous eigenvectors from the logical $ Z $ type operator. By adding the logical $ Z $ to the code stabilizer we can realize the logical states as stabilizer states. Every $ n $ qubit stabilizer state is proportional to $$ \sum_{u \in \mathbb{F}_2^k} i^{c^Ty} (-1)^{y^TQy} |y=Ru+t \rangle $$ for some vectors $ c,t \in \mathbb{F}_2^n $ and symmetric $ n \times n $ binary matrix $ Q $ and $ n \times k $ binary matrix $ R $. 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

It is interesting to note that many stabilizer states are graph states (and even if a stabilizer state is not explicitly a graph state then it is still equivalent by local Clifford gates to a graph state, see how to go from a stabilizer state to a graph ) and for the case of a graph state the matrix $ Q $ can be interpreted as the adjacency matrix of a a graph.

The relevant part of this equation is that a stabilizer state is either real (if $ c $ is perpendicular to the affine space $ R(\mathbb{F}_2^k)+t $ which is the support of the stabilizer state) or exactly half the support of the state has an imaginary coefficient (if $ c $ is not perpendicular to the support). In the case that the stabilizer state is half imaginary then it can be converted to a real stabilizer state by a local Clifford. Indeed the imaginary half will be exactly the cofactor of $ |0> $ or $ |1> $ for some qubit. If it is a cofactor of $ |0> $ then acting with $ XSX $ on that one qubit will take the state to a real state. Similarly, if the imaginary half is a cofactor of $ |1> $ then acting $ S $ on that qubit will take the state to a real state.

Answer 2

In this paper https://arxiv.org/abs/quant-ph/9611001 (Theorem 5) the author shows that "any additive code is equivalent to a real additive code". So you can always find an equivalent code where where all stabilizers are real; the codespace will then be spanned by combinations with $\pm 1$ coefficients only (no $\pm \imath$ needed). I always work with real codes for that reason (that's why I use $Y=XZ$ convention). I did notice that the number of nonzero coefficients is always a power of 2; nice to see that paper prove it.