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I am trying to understand the equivalence between the original planar surface code and the rotated version.

Looking at Tomita and Svore's paper (2014), they say

"The number of qubits [from original unrotated surface code] can be reduced while maintaining the same code distance by rotating it clockwise by 45 degrees and removing the four corner data qubits"

. My questions:

  • Is possible to make this notion more precise? (Why are we allowed to simply do this and be sure the pictures are equivalent?)

  • What procedure should one follow to remove the stabilizers (i.e. turning the weight-3s into weight-2s, and deleting some of them)?

  • In terms of their equivalence, how should I interpret the fact that the unrotated surface code has more stabilizers than the rotated one?

  • Besides being less resource intensive, are there any performance advantages of the rotated code?

I have been also checking some old papers from Bombin and Martin-Delgado (this one and this one) as well as Kovalev and Pryadko's work. While I appreciate the answers should be in there, I am failing distill a simple explanation from their discussions.

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