It is very common to look at stabilizer codes as codes over GF(2) or codes over GF(4). Mostly I have seen this for computations for distance of codes and syndromes. How do other notions like say automorphism group (over GF(2)) or some other invariants translate to stabilizer code? In other words will automorphism group of GF(2) equivalent code say something about its corresponding stabilizer code? For eg. Let $Q$ be a quantum code and corresponding GF(2) linear code be $C$ represented by generatir matrix (or parity check matrix M). Now given $aut(C)$ that is $\lbrace A,P \in GL(F_2) \times S_n \rbrace$ that fix M, what can we say about aut(Q), I suppose now automorphism groups would be defined over complex fields or considering stabilizer formalism we might have one over paulis if we want to look at stabilzers as analog of generator matrices.

  • $\begingroup$ Formulating them as codes over GF(2) or GF(4) is equivalent and helps in studying their structure. Of course, the properties of those GF(2)/GF(4) codes determine those of the stabiliser codes. Maybe you can make your question more precise, developing the formalism in an answer is beyond the scope of this site. Maybe, it would also help to read some of the literature, e.g. "Quantum Error Correction Via Codes Over GF(4)" by Calderbank, Rains, Shor and Sloane. $\endgroup$ Nov 16 '20 at 9:02
  • $\begingroup$ @Markus Heinrich see the edit. Let me know if I can add more things. $\endgroup$
    – Root
    Nov 16 '20 at 9:13
  • $\begingroup$ Maybe I'm missing the point. If $Q$ is a $[[n,k]]$ stabiliser code, then I would call the following its "corresponding $\mathbb{F}_2$ code: $Q$ projects on an isotropic linear subspace $C\subset\mathbb{F}_2^{2n}$ of dimension $n-k$. Here, isotropic means it is self-orthogonal w.r.t. the symplectic structure of $\mathbb{F}_2^{2n}$! If you want to change that for Euklidean orthogonality, you have to map it to an additive code over $\mathbb{F}_4$. With suitable definition, $\mathrm{Aut}(C)\subset\mathrm{Sp}_{2n}(\mathbb{F}_2)$ is a quotient of $\mathrm{Aut}(Q)$. Is it this what you mean? $\endgroup$ Nov 17 '20 at 8:52
  • $\begingroup$ @Markus Heinrich Yes. I was looking for 'suitable' defination and is there a way to go from $F_2$ linear (classical) code's aut group to the quantum analogue aut(C) to aut(Q). $\endgroup$
    – Root
    Nov 17 '20 at 9:18

Let me briefly elaborate on my comment. I'm not an expert on coding theory but I certainly know the symplectic formalism. If you want to know more about codes, I would recommend that you read "Quantum Error Correction Via Codes Over GF(4)" by Calderbank, Rains, Shor, and Sloane for the $\mathbb{F}_4$ formulation or "Quantum error-correcting codes and their geometries" by Ball, Centelles, and Huber for a $\mathbb F_2$ formulation.

I assume that you know that any element of the $n$-qubit Pauli group $P_n$ can be written uniquely as $W(z,x,t) = i^t Z(z)X(x)$ for $t\in\mathbb{Z}_4$, $z,x\in\mathbb{F}_2^{n}$ where $Z$ and $X$ are defined as usual as $Z(z)|u\rangle = (-1)^{z\cdot u}$, $X(x)|u\rangle = |u+x\rangle$. Note that the centre of $P_n$ is $\langle i \mathbb{I}\rangle$. We get a projection: $$ \pi : \; P_n \rightarrow \mathbb{F}_2^{2n}, \quad W(z,x,t) \mapsto (z,x). $$ Another way of seeing this is that $P_n$ is a group of order $2^{2n+2}$ which is "almost" extraspecial, i.e. the quotient $P_n/Z(P_n)$ is a (discrete symplectic) vector space. It is symplectic since $$ W(v,t)W(u,s) = (-1)^{[v,u]}W(u,s)W(v,s) $$ for $[v,u]:= v_z\cdot u_x - v_x \cdot u_z$ the standard symplectic product on $\mathbb F_2^{2n}$.

Since $\pi$ is a homomorphism, subgroups $S\subset P_n$ project to linear subspaces $\bar S:=\pi(S)\subset\mathbb F_2^{2n}$. By the above commutation relation, $S$ is Abelian if and only if $\bar S$ is isotropic, i.e. $S\subset S^\perp$ (self-orthogonal in coding language). Now it should be clear that any $[[n,k]]$ stabiliser code $Q$ given by a stabiliser group $S$ projects to an isotropic subspace $\bar S$. Since $S$ has rank $n-k$, the dimension of $\bar S$ is $n-k$, too.

A code word $s\in\bar S$ is of the form $s=(z_1,\dots,z_n,x_1,\dots,x_n)$. Its weight is defined as $\mathrm{wt}(s)=|\{(z_i,x_i)\neq 0\}|$. The automorphism group of $\bar S$ is then defined within the symplectic weight-preserving transformations on $\mathbb F_2^{2n}$. The latter are exactly the pair-wise coordinate permutations $S_n$ times the "local" symplectic group $\mathrm{Sp}_2(\mathbb F_2)$ acting on each symplectic pair independently, i.e. it has order $6^n n!$.

Now for the automorphism group of $Q$. Any symplectic map on $\mathbb F_2^{2n}$ is induced from a Clifford unitary and any two Cliffords induce the same map if and only if they differ by a Pauli operator. This means that $\mathrm{Aut}(\bar S)$ is basically $\mathrm{Aut}(Q)$ up to multiplication with Pauli operators. More precisely, $\mathrm{Aut}(\bar S) = \mathrm{Aut}(Q)/S$. The only difference between the two comes from the projection $\pi$.

Finally, let me remark that by mapping $(1,0) \mapsto 1$ and $(0,2)\mapsto \omega$ we get an additive mapping between $\mathbb F_2^{2n}$ and $\mathbb F_4^n$ which maps the isotropic linear codes in the above sense to self-orthogonal additive codes over $\mathbb F_4$. In this way, stabiliser codes fit better into the framework of coding theory, but this does not change the structure.


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