I am trying to better understand (standard, not gauge) 3d color codes. In particular, I am working with the lattice proposed in 1.

I understand how X error works, forming strings of the kind Vertex -> Cell -> Face -> Cell -> Vertex, where the Vertices and Faces are of the same color. (Notice that this is in the "dual picture", where qubits are at cells). An X error will happen if there is a string-net starting from one cell (qubit) and propagating to each face of the tetrahedra.

I would like to have a similar understanding of Z errors. From figure 2 in 2 it can be seen that Z logical errors propagate via the two-colored edges (we are in the dual picture, so faces -> edges) alternating edges of opposite coloring. For example red-yellow edge -> cell -> blue-green edge -> cell -> red-yellow edge.

What I am confused about is how Z logical errors look like. From 2 I would guess these have something to do with error branes encircling the face of the corresponding color, but I do not see it clearly.

My question is How do logical Z errors look like in this situation? Secondly, is it possible that I am getting it wrong that the lattice described in 1 and 3 can be used for non-gauge color codes?

Thanks a lot!

H Bombín, Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes

H. Bombin and M. A. Martin-Delgado, Topological Computation without Braiding

Benjamin J. Brown, Naomi H. Nickerson & Dan E. Browne Fault-tolerant error correction with the gauge color code

  • $\begingroup$ After thinking it a bit I believe that picture 4c in reference 2 above has a partial answer, it seems that there should be a string brane around each extreme vertex or face, depending on how you see it, and one of the colors of such string corresponds to the face or vertex. But the second question is still open. $\endgroup$
    – Pablo
    Jun 24, 2021 at 15:55


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