Use native implementation for adjoints in (control) operations

[josephleekl]

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---

Context:
Currently in _apply_lightning, we check for whether an operation is Adjoint, then we apply the operation with an adjoint (inv_param) flag. However, in cases where we have:

• adjoint(s) within control - e.g. control(adjoint(gate))
• control within adjoint - e.g. adjoint(control(gate)),

these are all applied as matrices.

Description of the Change:
_apply_lightning and _apply_lightning_controlled checks for adjoint in an operation, and if it's an adjoint it applies the base operation with an adjoint flag, instead of treating everything as a matrix.

So in effect we have:
control(adjoint(gate)) -> control(gate with adjoint)
adjoint(control(gate)) -> control(gate with adjoint)

which are implemented natively in C++ (if the gate is supported), yielding better performance

Benefits:
adjoint(ctrl()) will see the most speedup, especially with large number of control wires, since we use native control operation which contains less wires than the equivalent matrix, and needs to be operated on less wires. adjoint(ctrl()) will see some speed-up, since we are now able to use the native named gate implementation in C++.

Example timing improvement:
4 ctrl wires

LQ:

LQ, 25 qubits, 500 repeats	master	branch
ctrl(adjoint(IsingXX))	9.6s	6.0s
ctrl(adjoint(DoubleExcitationPlus))	27.6s	9.2s

LQ, 25 qubits, 100 repeats	master	branch
adjoint(ctrl(IsingXX))	267s	2.9s
adjoint(ctrl(DoubleExcitationPlus))	1002s	3.6s

Baseline:

LQ, 25 qubits, 500 repeats	master	branch
ctrl(IsingXX)	6.1s	6.1s
ctrl(DoubleExcitationPlus)	9.1s	9.1s

LG:

LG, 31 qubits, 1000 repeats	master	branch
ctrl(adjoint(IsingXX))	4.9s	4.8s
ctrl(adjoint(DoubleExcitationPlus))	5.0s	4.9s

LG, 31 qubits, 1000 repeats	master	branch
adjoint(ctrl(IsingXX))	119s	4.8s
adjoint(ctrl(DoubleExcitationPlus))	208s	4.9s

Baseline:

LG, 31 qubits, 1000 repeats	master	branch
ctrl(IsingXX)	4.8s	4.8s
ctrl(DoubleExcitationPlus)	4.9s	4.9s

LK:

LK, 25 qubits, 500 repeats	master	branch
ctrl(adjoint(IsingXX))	8.5s	5.7s
ctrl(adjoint(DoubleExcitationPlus))	24.5s	7.6s

LK, 25 qubits, 100 repeats	master	branch
adjoint(ctrl(IsingXX))	235s	2.6s
adjoint(ctrl(DoubleExcitationPlus))	867s	2.9s

Baseline:

LK, 25 qubits, 500 repeats	master	branch
ctrl(IsingXX)	5.6s	5.8s
ctrl(DoubleExcitationPlus)	7.7s	7.6 s

Possible Drawbacks:

Related GitHub Issues:

[sc-79430]

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[codecov]

Codecov Report

Attention: Patch coverage is 85.07463% with 10 lines in your changes missing coverage. Please review.

Project coverage is 97.35%. Comparing base (aad3e59) to head (a582803).
Report is 1 commits behind head on master.

Files with missing lines	Patch %	Lines
pennylane_lightning/lightning_gpu/_state_vector.py	9.09%	10 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1063      +/-   ##
==========================================
+ Coverage   96.10%   97.35%   +1.25%     
==========================================
  Files         232      111     -121     
  Lines       39178    16997   -22181     
==========================================
- Hits        37651    16547   -21104     
+ Misses       1527      450    -1077     

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[josephleekl]

Only LQ is implemented so far - but if this is OK, I'll propagate it to LK and LG, which is fairly straightforward
update: LK and LG added

[maliasadi]

Thanks @josephleekl!💡 Just a few comments before pushing it to master

[LuisAlfredoNu]

Thanks @josephleekl
Just a few comments ❄️

[maliasadi]

Nice work @josephleekl! Happy to approve 🎉

[LuisAlfredoNu]

Thank you @josephleekl

Happy to approve 🪂