Different codes with the same parameters

Question

The parameters of a code are $ [[n,k,d]] $ where $ n $ is the number of physical qubits, $ k $ is the number of logical qubits, and $ d $ is the distance.

There are lots of trivial codes with $ d=1 $ that do not detect or correct any errors. So in this question I am only interested in codes with at least $ d \geq 2 $, but preferably $ d \geq 3 $.

I will say that two codes are equivalent if they are related by local unitaries or by permutations. By local unitary I mean a unitary which is a tensor product of $ n $ single qubit unitaries (i.e. it acts locally on each qubit). For example, the Hadamard gate $H$ is a single qubit unitary. So $H^{\otimes n}$ would be a local unitary.

Question: What is an example of two non-equivalent codes that have the same parameters [[n,k,d]]? Again, I am restricting here to at least $ d \geq 2 $, preferably $ d \geq 3 $.

Answer 1

Example 1

The $3\times 3$ rotated surface code and the $9$-qubit Shor's code are both $[\![9,1,3]\!]$ codes.

They are inequivalent under local unitaries since the logical states have different entanglement structure. For example, the logical computational basis states of the $9$-qubit Shor's code have no entanglement between the first three qubits and the other six. By contrast, any three qubits in the surface code are entangled with the other six$^1$.

Example 2

Start with the $[\![7,1,3]\!]$ Steane code, add two physical qubits and just let them sit in the fridge (or wherever you keep your qubits). Encoding, decoding and logical operators ignore the two qubits, i.e. act on them as identity. This is another $[\![9,1,3]\!]$ code.

This code is inequivalent to either of the two codes in example 1 since now all logical states (not just the logical computational basis states) lack entanglement between any of the idle qubits and the remaining seven.

More examples

We can construct further examples using the idle qubits trick, e.g. add four idles to the $[\![5,1,3]\!]$ code, or by concatenation, e.g. concatenating $[\![5,1,3]\!]$ on top of $[\![9,1,3]\!]$ yields a code inequivalent to the one obtained by concatenating $[\![9,1,3]\!]$ on top of $[\![5,1,3]\!]$ as can be shown using entanglement structure arguments like those above.

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^{$^1$ Perhaps the easiest way to see this is to note that if we loose any six qubits of a $3\times 3$ surface code patch then we have lost the ability to measure one or both of the $X$ and $Z$ observables and thus have essentially lost quantum information encoded in the code. This is not necessarily the case for the $9$-qubit Shor's code: carefully chosen three qubits preserve full quantum information encoded in the code.}