Fix the calculation for identity operator's expectation value (#13345)

* Fix the calculation for identity operator's expectation value

* Add release note and test

* reformatted `test_statevector.py` with black

* Update releasenotes/notes/fix_identity_operator_9e2ec9770ac046a6.yaml

Co-authored-by: Julien Gacon <gaconju@gmail.com>

* add a docstring to pass the pylint test

* Fix leading `.`

---------

Co-authored-by: Julien Gacon <gaconju@gmail.com>

qiskit/quantum_info/states/statevector.py

@@ -476,7 +476,7 @@ def _expectation_value_pauli(self, pauli, qargs=None):
         pauli_phase = (-1j) ** pauli.phase if pauli.phase else 1
 
         if x_mask + z_mask == 0:
-            return pauli_phase * np.linalg.norm(self.data)
+            return pauli_phase * np.linalg.norm(self.data) ** 2
 
         if x_mask == 0:
             return pauli_phase * expval_pauli_no_x(self.data, self.num_qubits, z_mask)

releasenotes/notes/fix_identity_operator_9e2ec9770ac046a6.yaml

@@ -0,0 +1,5 @@
+---
+fixes:
+  - |
+    Fixed a bug that caused :meth:`.Statevector.expectation_value` to yield incorrect results
+    for the identity operator when the statevector was not normalized.

test/python/quantum_info/states/test_statevector.py

@@ -1152,6 +1152,31 @@ def test_expval_pauli_qargs(self, qubits):
         expval = state.expectation_value(op, qubits)
         self.assertAlmostEqual(expval, target)
 
+    def test_expval_identity(self):
+        """Test whether the calculation for identity operator has been fixed"""
+
+        # 1 qubit case test
+        state_1 = Statevector.from_label("0")
+        state_1_n1 = 2 * state_1  # test the same state with different norms
+        state_1_n2 = (1 + 2j) * state_1
+        identity_op_1 = SparsePauliOp.from_list([("I", 1)])
+        expval_state_1 = state_1.expectation_value(identity_op_1)
+        expval_state_1_n1 = state_1_n1.expectation_value(identity_op_1)
+        expval_state_1_n2 = state_1_n2.expectation_value(identity_op_1)
+        self.assertAlmostEqual(expval_state_1, 1.0 + 0j)
+        self.assertAlmostEqual(expval_state_1_n1, 4 + 0j)
+        self.assertAlmostEqual(expval_state_1_n2, 5 + 0j)
+
+        # Let's try a multi-qubit case
+        n_qubits = 3
+        state_coeff = 3 - 4j
+        op_coeff = 2 - 2j
+        state_test = state_coeff * Statevector.from_label("0" * n_qubits)
+        op_test = SparsePauliOp.from_list([("I" * n_qubits, op_coeff)])
+        expval = state_test.expectation_value(op_test)
+        target = op_coeff * np.abs(state_coeff) ** 2
+        self.assertAlmostEqual(expval, target)
+
     @data(*(qargs for i in range(4) for qargs in permutations(range(4), r=i + 1)))
     def test_probabilities_qargs(self, qargs):
         """Test probabilities method with qargs"""