How do Union-Find Decoders deal with Measurement errors through multiple measurement rounds?

Question

I've read a few papers regarding to Surface Code and its decoding algorithms. I've learned that a Union-Find decoder need up to $d$ measurement rounds to deal with measurement errors.

These measurement rounds yields a 3-D syndrome graph, like this:

I tried to understand the way Union-Find Decoders deal with the 3D syndrome graphs but failed. I haven't found a paper or open-source implementation which clarify the foundamental steps of the decoder.

So here is my question:

1. How to understand the way Union-Find Decoders deal with multiple measurement rounds and 3D syndrome graphs?
2. Is there any papers or open-source implementations of this measurement-error-solvable decoder?

If you could provide any guidance or insights on this matter, I would be immensely grateful. Your expertise would be invaluable in helping me understand this topic. Thank you in advance for your time and knowledge.

Answer 1

It doesn't do anything special at all to deal with the measurement errors. The circuit noise problem is conceptually identical to the code capacity noise problem. Errors cause 2 or 1 detection events (defining the edges and boundary edges of a graph) and you match up the detection events by growing search regions on that graph. Literally the exact same code works when you add measurement errors.