Improve `__pow__` for `SingleQubitCliffordGate` and `CliffordGate` class

[NoureldinYosri]

Is your feature request related to a use case or problem? Please describe.
The __pow__ operator for CliffordGate is implemented only for integer powers and has complexity $\mathcal{O}(n)$ where $n$ is the exponent.

Cirq/cirq-core/cirq/ops/clifford_gate.py

Lines 399 to 411 in ec84a05

	def __pow__(self, exponent) -> 'CliffordGate':
	if exponent == -1:
	return CliffordGate.from_clifford_tableau(self.clifford_tableau.inverse())
	if exponent > 0 and int(exponent) == exponent:
	base_tableau = self.clifford_tableau.copy()
	for _ in range(int(exponent) - 1):
	base_tableau = base_tableau.then(self.clifford_tableau)
	return CliffordGate.from_clifford_tableau(base_tableau)
	if exponent < 0 and int(exponent) == exponent:
	base_tableau = self.clifford_tableau.copy()
	for _ in range(int(-exponent) - 1):
	base_tableau = base_tableau.then(self.clifford_tableau)
	return CliffordGate.from_clifford_tableau(base_tableau.inverse())

For SingleQubitCliffordGate it's implemented for only integer powers where it falls to CliffordGate.__pow__ and for $\pm \sqrt{}$.

Cirq/cirq-core/cirq/ops/clifford_gate.py

Lines 718 to 728 in ec84a05

	def __pow__(self, exponent) -> 'SingleQubitCliffordGate':
	# First to check if we can get the sqrt and negative sqrt Clifford.
	if self._get_sqrt_map().get(exponent, None):
	pow_gate = self._get_sqrt_map()[exponent].get(self, None)
	if pow_gate:
	return pow_gate
	# If not, we try the Clifford Tableau based method.
	ret_gate = super().__pow__(exponent)
	if ret_gate is NotImplemented:
	return NotImplemented
	return SingleQubitCliffordGate.from_clifford_tableau(ret_gate.clifford_tableau)

Describe the solution you'd like
For CliffordGate.__pow__ exponentiation should be done using binary exponentiation to reduce the complexity ot $\mathcal{O}(\log{n})$. Support for non integer exponents is hard in the general case.

For SingleQubitCliffordGate.__pow__. The single qubit clifford gates are a group of size 24. see.

Cirq/cirq-core/cirq/ops/clifford_gate.py

Line 149 in ec84a05

def all_single_qubit_cliffords(cls) -> Sequence['cirq.SingleQubitCliffordGate']:

support for integer powers can be done in $\mathcal{O}(1)$ if we either fall to the optimized CliffordGate.__pow__ but with exponent%24 instead of expnent or cache the results in table and access group_powers[self][exponent%24]. For rational exponents, When the clifford operation has a sqrt. The operation becomes well defined for exponents of the form $\frac{k}{2}$ where $k \in \mathbb{Z}$. For example $X^\frac{5}{2}$ is the same as $SqrtX^5$ and $X^\frac{-5}{2}$ which is the same as $(SqrtX^\dagger)^5$.

What is the urgency from your perspective for this issue? Is it blocking important work?

P3 - I'm not really blocked by it, it is an idea I'd like to discuss / suggestion based on principle

[suyashdamle]

I'd like to take this one up

[telescopic]

@tanujkhattar - Is this issue still open and not actively being worked on? I'd be happy to take it up :)

( Slightly tangential question - looking at some of the other bugs, should assignees be un-assigned after some staleness limit? I am new to this repo and wanted to see what would issues would be available basis presence/absence of assignees )

[suyashdamle]

I'm trying to find time to put into this. If its a high priority issue, feel free to reassign, if not, I'll get to it this week

[telescopic]

Gotcha, lmk if you'd like to pass it on, thanks!

[migueltorrescosta]

Should this be reassigned?

[NoureldinYosri]

yes, I think this issue has been inactive long enough, @migueltorrescosta I reassigned it to you