Memoize compile_decomposition_rules per operator variant

[Qottmann]

Context:
Closes #3173.

Description of the Change:
Under qml.qjit(capture=True) graph decomposition, the compiler's on-demand rule loader rebuilds the full reachable-rule closure for every operator it queries. As a result, compile_decomposition_rules recompiles the same operator variant many times over — on the RZ phase-gradient demo it was invoked ~5870 times for only ~179 unique variants, making rule compilation the dominant cost of decomposition-heavy programs.

compile_decomposition_rules is a pure function of its arguments, so this PR memoizes its result in a module-level cache (_COMPILE_DECOMP_CACHE) keyed by the fully-resolved operator instance and modifier variant (op_name, op_id, dynamic_shape, wire_lens, static_data, extra_data, is_custom_op, wrap_adjoint, wrap_control, n_ctrl). Each unique variant is now compiled once.

Correctness: the cached inlined_module is only ever read downstream (walked and cloned), never mutated in place, so sharing the same module across callers is safe.

Benefits:
Eliminates redundant recompilation of identical decomposition-rule variants, substantially reducing compile time for decomposition-heavy programs under capture.

Possible Drawbacks:
The cache persists for the process lifetime. Values are keyed by a stable, hashable representation of the fully-resolved operator variant, so distinct variants remain distinct.

Related GitHub Issues:
Closes #3173.

Made with Cursor

[codecov]

Codecov Report

❌ Patch coverage is 89.47368% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 95.64%. Comparing base (d408f1f) to head (94bba5d).
⚠️ Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
...tend/catalyst/decomposition/decomposition_rules.py	89.47%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3174      +/-   ##
==========================================
- Coverage   95.65%   95.64%   -0.01%     
==========================================
  Files         177      177              
  Lines       20763    20782      +19     
  Branches     2080     2083       +3     
==========================================
+ Hits        19860    19877      +17     
- Misses        718      720       +2     
  Partials      185      185              

☔ View full report in Codecov by Harness.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.