Update and correct pow function for PauliSum (#6019)

* Update and correct pow function for PauliSum

Updated pow function for PauliSum to use binary exponentiation

* Format and remainder initialisation correction

+Corrected formatting
+Corrected initialisation of remainder to identity

* Format and remainder initialisation correction

+Corrected formatting
+Corrected initialisation of remainder to identity

* using local variable instead of self and reformatting

* reverting to regular linear multiplication and adding tests

* Adding test for range (1,9)

* trailing blank

* remove trailing blank test

* Getting rid of extraneous variables

* Fix up final formatting issue

---------

Co-authored-by: Tanuj Khattar <tanujkhattar@google.com>
Co-authored-by: Pavol Juhas <juhas@google.com>

cirq-core/cirq/ops/linear_combinations.py

@@ -839,10 +839,10 @@ def __pow__(self, exponent: int):
         if exponent == 0:
             return PauliSum(value.LinearDict({frozenset(): 1 + 0j}))
         if exponent > 0:
-            base = self.copy()
+            result = self.copy()
             for _ in range(exponent - 1):
-                base *= base
-            return base
+                result *= self
+            return result
         return NotImplemented
 
     def __truediv__(self, a: value.Scalar):

cirq-core/cirq/ops/linear_combinations_test.py

@@ -1156,6 +1156,22 @@ def test_pauli_sum_pow():
     for psum in [psum1, psum2, psum3, psum4]:
         assert cirq.approx_eq(psum**0, identity)
 
+    # tests for exponents greater than two for both even and odd
+    psum5 = cirq.Z(q0) * cirq.Z(q1) + cirq.Z(q2) + cirq.Z(q3)
+    correctresult = psum5.copy()
+    for e in range(1, 9):
+        assert correctresult == psum5**e
+        correctresult *= psum5
+
+    psum6 = cirq.X(q0) * cirq.Y(q1) + cirq.Z(q2) + cirq.X(q3)
+    assert psum6 * psum6 * psum6 * psum6 * psum6 * psum6 * psum6 * psum6 == psum6**8
+
+    # test to ensure pow doesn't make any change to the original value
+    psum7 = cirq.X(q0) * cirq.Y(q1) + cirq.Z(q2)
+    psum7copy = psum7.copy()
+    assert psum7**5 == psum7 * psum7 * psum7 * psum7 * psum7
+    assert psum7copy == psum7
+
 
 # Using the entries of table 1 of https://arxiv.org/abs/1804.09130 as golden values.
 @pytest.mark.parametrize(