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Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault-tolerantly.
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To prevent errors from accumulating, error correction needs to be applied after each encoded gate. We implicitly assume that error correction itself is fault-tolerant for any TSC due to locality of the check operators
Cited in the paper.
Any operator U ∈ 𝒫 D U\in{\cal P}_{D} is specified (up to an overall phase) by a list of 2 k 2k operators { U X j U † , U Z j U † ∈ 𝒫 D − 1 } \{UX_{j}U^{\dagger},UZ_{j}U^{\dagger}\in{\cal P}_{D-1}\} , where j = 1 , … , k j=1,\ldots,k . This shows that | 𝒫 D | ≤ | 𝒫 D − 1 | 2 k ≤ 2 ( 2 k ) D |{\cal P}_{D}|\leq|{\cal P}_{D-1}|^{2k}\leq 2^{(2k)^{D}}
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Our notation is different from Ref. [ 13 ]
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H. Bombin, arXiv:1107.2707 (2011)
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