Line label reordering during `Circuit` concatenation

[sserita]

Line labels can potentially be reordered during + between Circuit objects. Some protocols (such as circuit-mirroring-based techniques) compute expected output bitstrings, and this addition reordering can happen unexpectedly and cause a mismatch between final circuit line labels and implicit line labels of the computed bitstring.

We're catching these reorderings in our MCFE-based code now, but we should revisit this to see if there could have been better behavior to prevent the bug in the first place.

[sserita]

@coreyostrove I seem to recall you having a counterpoint to this being an issue we should address directly. Do you remember what that was? I think you were mid-Circuit overhaul at the time.

[coreyostrove]

I think my objection was primarily to this being a bug, per se. The reordering/normalization of the line labels was an intentional design choice. Whether it was a good one is another story.

Obviously this gave rise to a bug in the context of MCFE, but it keeping the line labels in their original order would also introduce some bugs down the line (and probably upon making the change). I think it'd be worth discussing what properties we want the concatenation of circuits to have. The desiderata that drove the re-ordering was that I wanted the line labeling to match the "intuitive commutativity" that someone would use if they were thinking about this in a pen-and-paper/white board setting. In that case I think that if I asked someone to concatenate a circuit with support on a qubit labeled "Q1" and a circuit with support on a qubit labeled "Q2" and then asked them to state what the list of qubit labels for the composition was they'd return ('Q1', 'Q2') independent of which of the two circuits I handed them first (and similarly draw the composition in an ordered fashion even if the original labels were out of order). There are some obvious edge cases in that heuristic (suppose I gave you "Qcricket" and "Qbaseball", here I think the ordering may reflect whether you think sports are boring or not, and if not would vary in Japan and Australia...).