Improve approximate set generation in qml.ops.sk_decomposition

[obliviateandsurrender]

Context: The approximate set for the Solovay-Kitaev decomposition is currently prepared by populating a trie-based data structure (TbDS) with a BFS traversal. This is efficient but misses out on a few future gate sequences because some parent sequences are caught in the equality check condition.

Description of the Change:

1. To allow traversal over some potential missed-out candidate sequences, we relax the equality check to include information regarding the tracked global phases. This unlocks.
2. While the previous step unlocks the previously forbidden parent sequences, it also adds redundant sequences in the set, which is undesirable. So we prune our TbDS by finding the redundant pool sequences and ranking them based on their T-gate count.
3. To perform the T-gate ranking efficiently, we also build and store this information for each sequence during the traversal itself.
4. To unlock higher max_length for broader TbDS, we now separate checking for equivalence with a local and global KDTree. The local one is updated as the TbDS is being constructed for a given depth, whereas the global one is updated after the construction for a given depth.

Benefits: This makes the approximate set decently dense than before, unlocking efficient decomposition for some previously missing edge cases.

Possible Drawbacks: This adds a constant overhead incurred from traversing additional parent nodes and post-build pruning. However, this is negligible; as the approximate set once built is cached.

Related GitHub Issues: [sc-82463] #6131

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.59%. Comparing base (656dedc) to head (19b2452).
Report is 4 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #6855      +/-   ##
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- Coverage   99.59%   99.59%   -0.01%     
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  Files         478      478              
  Lines       45343    45333      -10     
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- Hits        45161    45151      -10     
  Misses        182      182              

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[isaacdevlugt]

Thanks @obliviateandsurrender!

[willjmax]

Mostly just concerns with readability.

[austingmhuang]

Thanks @obliviateandsurrender. Wonder if there is a better solution than this local/global trie thing going on though.

[willjmax]

I think the local/global stuff is still a little hard to read, but the added comments should make future maintenance less painful.