Are there any implications of the automorphism group in QECC?

Question

We often see that classically automorphism group of an error-correcting code plays a crucial role in many computational problems. Are there any important implications that depend on this in quantum case?

One I could find was this fault tolerant computing. In that case automorphism group for underlying codes to CSS codes are mainly considered.

I am looking for more such applications of automorphism groups of QECC. Also in the case of stabilizer codes, does the automorphisms group have any direct relation to its stabilizer group, and is there a standard definition of "automorphism group" of relavance?

Answer 1

In the case of stabilizer codes, one generally starts with the group, $\cal{G}$, of tensor products of basis vectors on $n$ qubits. On one qubit the applicable group is the Pauli Group, which is order 16, call it $\cal{G}_0$. So at the general level $\cal{G} = \bigotimes_{i=1}^n \cal{G}_0$. Simplifying assumptions are made in many treatments that make it difficult (for me at least) to pin down the exact discrete groups, and irreps of those groups, being invoked for different stabilizer code applications.

The stabilizer subgroup, $\cal{S} < \cal{G}$, is an Abelian subgroup of $\cal{G}$ that fixes the codespace, $\mathbf{T}$. Note that $\mathbf{T}$ doesn't necessarily have a group structure, so it's a subspace of $\cal{G}$. Since $\mathbf{T}$ is the space of vectors fixed by $\cal{S}$, the action of $\cal{S}$ on $\mathbf{T}$ is the trivial automorphism of $\mathbf{T}$ (i.e. the identity).

As indicated by the name, it's generally the structure of $\cal{S}$ and $\text{Aut}(\cal{S})$ that are most interesting and useful. $\text{Aut}(\cal{S})$ determines the set of valid fault-tolerant encoded operations, so codes with large $\text{Aut}(\cal{S})$ are desireable. Since $\cal{S}$ is Abelian there are no non-trivial inner automorphisms of $\cal{S}$, and $\text{Aut}({\cal{S}})=\text{Out}(\cal{S})$.

So at least in the general theory, the most interesting automorphisms are the trivial automorphisms of the codespace, which defines the stabilizer subgroup, and the outer automorphisms of the stabilizer subgroup, which enable fault tolerant operations. The best reference for all of this that I have found is Gottesman's Thesis, which reads like a textbook on the subject.

As a final note, conventional stabilizer QECC's are a special case of stabilizer operator QECC's. In the context of OQECC's gauge symmetries are exploited to make codes more efficient, so the normalizer of $\cal{S}$ plays an important role. The standard reference for OQECC from Poulin is also very helpful in understanding the group structure of conventional stabilizer codes.