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Two quantum codes are considered equivalent if they are related by local unitarys and permutations, i.e., if they are related by non-entangling gates.

Given two codes, is there an easy way to tell that they are not equivalent?

I already know one can use quantum weight enumerators, i.e., if the weight enumerators of each code are different then the codes must be inequivalent. However, if the two codes have the same weight enumerators they still could be inequivalent. For example, in https://arxiv.org/pdf/0709.1780.pdf, the authors show that there are 16 inequivalent $[[7,1,3]]$ stabilizer codes but they correspond to only 10 weight enumerators.

Is there an alternative, easy to compute, code invariant?

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It turns out that in that same paper, https://arxiv.org/pdf/0709.1780.pdf, there is a method called "frequency analysis." The idea is to calculate the weight enumerators while keeping track of which qubits the Pauli operators act on. In this sense, it is a "finer" weight enumerator.

One can check that this fine weight enumerator is invariant under local unitarys and permutations (as long as we fix an ordering).

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