Optimize Clifford.__pow__ by using binary exponentiation (#6581)

cirq-core/cirq/ops/clifford_gate.py

@@ -400,7 +400,11 @@ def _has_stabilizer_effect_(self) -> Optional[bool]:
         # By definition, Clifford Gate should always return True.
         return True
 
-    def __pow__(self, exponent) -> 'CliffordGate':
+    def __pow__(self, exponent: float) -> 'CliffordGate':
+        if exponent != int(exponent):
+            return NotImplemented
+        exponent = int(exponent)
+
         if exponent == -1:
             return CliffordGate.from_clifford_tableau(self.clifford_tableau.inverse())
         if exponent == 0:
@@ -409,18 +413,24 @@ def __pow__(self, exponent) -> 'CliffordGate':
             )
         if exponent == 1:
             return self
-        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())
 
-        return NotImplemented
+        base_tableau = self.clifford_tableau.copy()
+        if exponent < 0:
+            base_tableau = base_tableau.inverse()
+            exponent = abs(exponent)
+
+        # https://cp-algorithms.com/algebra/binary-exp.html
+        aux = qis.CliffordTableau(
+            num_qubits=self.clifford_tableau.n
+        )  # this tableau collects the odd terms
+        while exponent > 1:
+            if exponent & 1:
+                aux = aux.then(base_tableau)
+            base_tableau = base_tableau.then(base_tableau)
+            exponent >>= 1
+
+        base_tableau = base_tableau.then(aux)
+        return CliffordGate.from_clifford_tableau(base_tableau)
 
     def __repr__(self) -> str:
         return f"Clifford Gate with Tableau:\n {self.clifford_tableau._str_full_()}"

cirq-core/cirq/qis/clifford_tableau.py

@@ -129,7 +129,9 @@ def apply_global_phase(self, coefficient: linear_dict.Scalar):
 
 class CliffordTableau(StabilizerState):
     """Tableau representation of a stabilizer state
-    (based on Aaronson and Gottesman 2006).
+
+    References:
+        - [Aaronson and Gottesman](https://arxiv.org/abs/quant-ph/0406196)
 
     The tableau stores the stabilizer generators of
     the state using three binary arrays: xs, zs, and rs.