Optimize __pow__ for CliffordGate (Issue #6327)

[Vivek1106-04]

Description:
This PR addresses Issue #6327 by optimizing the __pow__ method for both CliffordGate and SingleQubitCliffordGate.

Changes proposed in this PR:

1. CliffordGate Optimization:

   • Replaced the $O(N)$ linear loop with Binary Exponentiation (exponentiation by squaring).
   • This reduces the complexity to $O(\log N)$, significantly improving performance for large exponents.
   • Added a safety check to ensure result_tableau is initialized independently to avoid reference aliasing bugs.

2. SingleQubitCliffordGate Optimization:

   • Integer Powers: Leveraged the group structure of single-qubit Cliffords (size 24). All integer exponents are now reduced modulo 24 before calculation.
   • Rational Powers: Added support for half-integer exponents (e.g., gate**2.5). The logic decomposes $G^{k/2}$ into $(G^{0.5})^k$ and recursively calls the optimized integer power function.
   • Added type checking to ensure invalid inputs (like strings) return NotImplemented.

Related Issue:
Fixes #6327

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 99.57%. Comparing base (4d75b4b) to head (c127a01).

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #7766   +/-   ##
=======================================
  Coverage   99.57%   99.57%           
=======================================
  Files        1102     1102           
  Lines       98434    98436    +2     
=======================================
+ Hits        98014    98016    +2     
  Misses        420      420           

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[pavoljuhas]

@codrut3 - can you please take a look at this? (If so, please post short comment so I can assign you as a reviewer)

[mhucka]

Discussed in Cirq Cynq: #6919 had implemented a solution for the SingleQubitCliffordGate case. Suggestion: split this PR into 2 PRs,

1. Modify this current PR to be just about the CliffordGate part (which is different from Improve __pow__ for SingleQubitCliffordGate (issue #6327) #6919)
2. Do a new PR for the enhancements to the SingleQubitCliffordGate optimization case

[codrut3]

@codrut3 - can you please take a look at this? (If so, please post short comment so I can assign you as a reviewer)

Happy to review! Thanks Pavol!

[codrut3]

Hey @Vivek1106-04 ,
Looking at the existing code, both optimizations requested by #6327 are already implemented. Your PR is re-arranging if order in some cases and adding explanatory comments, but not changing the logic. In particular, the exponentiation is already done in $O(\log N)$ steps. You are also removing the base case if exponent == -1: which, although still correct, seems less efficient to me.

The only new improvement is on line 413, where you are using self.clifford_tableau.inverse() directly if the exponent is $< 0$, instead of first making a copy of the tableau (the inverse() method creates a new tableau so this is safe).
Perhaps this line change can be submitted separately as a PR with an explanatory description while leaving the rest of the code unchanged?

@pavoljuhas #6327 is already solved and should be closed to avoid confusion.