Example stabilizer code which is not $ GF(4) $ linear

Question

In the paper

https://arxiv.org/abs/quant-ph/9704043

Eric Rains talks about $ GF(4) $ linear codes and proves some of their properties, for example

• "many codes of interest (e.g., GF(4)-linear codes) are built out of distance 2 codes."

• "Since GF(4)-linear codes are built out of [[2n, 2(n−1), 2]]s, we find, for instance, that any equivalence between GF(4)-linear codes (subject to certain trivial restrictions) must lie in the Clifford group."

• "Corollary 14. Any equivalence of GF(4)-linear quantum codes lies in the Clifford group, unless the codes have minimum distance 1, or contain a codeword of weight 2."

• "Lemma 15. A GF(4)-linear code C is spanned by its minimal codewords."

• "Corollary 16. If Q is a GF(4)-linear code, then every automorphism of Q lies in the Clifford group."

I'm trying to better understand the limited scope of these results.

What is an example of a stabilizer code which is not $ GF(4) $ linear?

Update: The comment from @unknown says that a CSS code is $ GF(4) $ linear if and only if $ H_x=H_z $. Now I'm curious if the $ [[5,1,3]] $ code is $ GF(4) $ linear.

Answer 1

For $[[5,1,3]]$ code :

stabilizers : [0,1,0,0,1, 0,0,1,1,0] ~ IXZZX

[[0,1,0,0,1,0,0,1,1,0],
 [1,0,1,0,0,0,0,0,1,1],
 [0,1,0,1,0,1,0,0,0,1],
 [0,0,1,0,1,1,1,0,0,0]];

GF(4) form : [0,1,2,3] ~ [0,1,w,w^2]

[[0,1,2,2,1],
 [1,0,1,2,2],
 [2,1,0,1,2],
 [2,2,1,0,1]]

the first matrix has GF(2) rank = 4; code size = 2^4 = 16

the second has GF(4) rank = 2 ; code size = 4^2=16

so the code is linear.

For $[[15,1,3]]$ RM code the two matrices are :

[[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1]];
[[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],
 [0,1,1,0,0,1,1,0,0,1,1,0,0,1,1],
 [0,0,0,1,1,1,1,0,0,0,0,1,1,1,1],
 [0,0,0,0,0,0,0,1,1,1,1,1,1,1,1],
 [2,0,2,0,2,0,2,0,2,0,2,0,2,0,2],
 [0,2,2,0,0,2,2,0,0,2,2,0,0,2,2],
 [0,0,0,2,2,2,2,0,0,0,0,2,2,2,2],
 [0,0,0,0,0,0,0,2,2,2,2,2,2,2,2],
 [0,0,2,0,0,0,2,0,0,0,2,0,0,0,2],
 [0,0,0,0,2,0,2,0,0,0,0,0,2,0,2],
 [0,0,0,0,0,2,2,0,0,0,0,0,0,2,2],
 [0,0,0,0,0,0,0,0,2,0,2,0,2,0,2],
 [0,0,0,0,0,0,0,0,0,2,2,0,0,2,2],
 [0,0,0,0,0,0,0,0,0,0,0,2,2,2,2]];

the first matrix has GF(2) rank = 14; code size = 2^14 = 16384

the second has GF(4) rank = 10 ; code size = 4^10 = 1048576

so the code is not linear.

Answer 2

Theorem 4 in https://arxiv.org/pdf/quant-ph/9608006.pdf says that linear $GF(4)$ codes are even. In particular this means that odd codes cannot be linear.

As an example, consider the $[[5,1,2]]$ CSS code given by stabilizer generators \begin{align*} ZZZZZ \\ XXXXI \\ IXXXX \\ XXIXX. \end{align*} This code is odd because the first generator has weight $5$. Note that one can easily check that there are no weight $1$ elements in the stabilizer.

If we use $I \to 0$, $Z \to 1$, $X \to \omega$, and $Y \to \omega^2$, then the stabilizer in $GF(4)$ is \begin{align*} 11111 \\ \omega \omega \omega \omega 0 \\ 0 \omega \omega \omega \omega \\ \omega \omega 0 \omega \omega. \end{align*} If we multiply the second generator by $\omega$ twice and use the fact that $\omega^3 = 1$ then we get the element $11110$. But that added with the first generator is $00001$. But we know there are no weight 1 elements in the stabilizer so this code is not closed under $\omega$ and hence it is not linear (as expected since it is odd).

More generally, for linear codes if $g$ is in the stabilizer then $\omega g$ and $\omega^2 g$ must also be in the stabilizer. But this is a sort of "Pauli cycle", e.g., $\omega g$ is just $g$ with each $X$ replaced by $Y$, each $Y$ replaced by $Z$, each $Z$ replaced by $X$ and each $I$ left alone (and $\omega^2 g$ is the same except in reverse order).

For example, the $[[5,1,3]]$ code has generators $IIIII$ and \begin{align*} (XZZXI)_\text{cyc} \\ (YXXYI)_\text{cyc} \\ (ZYYZI)_\text{cyc} \end{align*} where the subscript indicates all 5 cyclic shifts occur. This way of writing the stabilizer makes it clear that the code is linear because the 2nd line is just $\omega$ times the first and the 3rd line is just $\omega^2$ times the first.

Thus a linear $GF(4)$ code is a stabilizer code that is even and whose stabilizer is closed under "Pauli cycles".

Note there is also a restriction on $n$ and $k$. For an $[[n,k,d]]$ code, the stabilizer has size $2^{n-k}$. On the other hand, for linear codes, the size must be $1 + 3 p$ for some integer $p$. To be sure, the $+1$ comes from the identity and the $3p$ comes because each generator comes in a package of $3$ from the "Pauli cycles." Thus we must have $1+3p = 2^{n-k}$ or $p = (2^{n-k}-1)/3$. However, $p$ is an integer iff $n-k$ is even. It follows that linear codes must have $n-k$ even.

Then for example, linear stabilizer states $[[n,0,d]]$ can only occur when $n$ is even. Moreover, $[[n,1,d]]$ can only occur for $n$ odd.