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I was reading combinatorial approach towards quantum correction. A lot of work in this is on finding diagonal distance of a graph. Let me add definition of diagonal distance so that this remains self-contained.

Given a labeling $L$ (a map where each vertex is is assigned 1 or 0) we define two operations on this:

  1. $X(v,L)$: you flip labeling of vertex v that is if it was zero make it 1 if it was 1 make it 0.

  2. $Z(v,L)$: you flip labeling of every neighbor of vertex v

Then diagonal distance is defined as length of minimal non-trivial sequence of operation so that $L$ is taken back to itself.

How is this exactly related to the quantum error correction property?

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