Added support for non-hermitian operators in AerPauliExpectation

Fixes Qiskit#415

QEOM creates a dictionary of operators to evaluate on the groundstate.
When using the noisy simulators (qasm_simulator or aer_simulator), one
could either use PauliExpectation (with noise) or AerPauliExpectation
(without noise). PauliExpectation works with non-hermitian operators
but internal methods of AerPauliExpectation raised an Error.
This is a workaround to this limitation.
Note that using include_custom=True on qasm allows the VQE to use a
local AerPauliExpectation without using the "expectation" input.
This does not apply to QEOM and one should explicitly define the
"expectation" input of the VQE for it to apply globally.

qiskit/opflow/expectations/aer_pauli_expectation.py

@@ -43,13 +43,31 @@ def convert(self, operator: OperatorBase) -> OperatorBase:
         AerSnapshot-based expectation circuits.
 
         Args:
-            operator: The operator to convert.
+            operator: The operator to convert. If it contains non-hermitian terms, apply an additional step.
 
         Returns:
             The converted operator.
         """
+
         if isinstance(operator, OperatorStateFn) and operator.is_measurement:
-            return self._replace_pauli_sums(operator.primitive) * operator.coeff
+            is_herm = operator.primitive.is_hermitian()
+            if not is_herm:
+                pauli_sum_re = (
+                    self._replace_pauli_sums(
+                        1 / 2 * (operator.primitive + operator.primitive.adjoint()).reduce()
+                    )
+                    * operator.coeff
+                )
+                pauli_sum_im = (
+                    self._replace_pauli_sums(
+                        1 / 2j * (operator.primitive - operator.primitive.adjoint()).reduce()
+                    )
+                    * operator.coeff
+                )
+                pauli_sum = (pauli_sum_re + 1j * pauli_sum_im).reduce()
+            else:
+                pauli_sum = self._replace_pauli_sums(operator.primitive) * operator.coeff
+            return pauli_sum
         elif isinstance(operator, ListOp):
             return operator.traverse(self.convert)
         else: