Make primitives's key serializable (#8796)

* serializable key

* refactor (Imamichi-san's suggestion)

* refactoring (Imamichi-san's suggestion)

Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>

qiskit/primitives/utils.py

@@ -16,6 +16,7 @@
 from __future__ import annotations
 
 from qiskit.circuit import Instruction, ParameterExpression, QuantumCircuit
+from qiskit.circuit.bit import Bit
 from qiskit.extensions.quantum_initializer.initializer import Initialize
 from qiskit.opflow import PauliSumOp
 from qiskit.quantum_info import SparsePauliOp, Statevector
@@ -113,6 +114,16 @@ def final_measurement_mapping(circuit: QuantumCircuit) -> dict[int, int]:
     return mapping
 
 
+def _bits_key(bits: tuple[Bit, ...], circuit: QuantumCircuit) -> tuple:
+    return tuple(
+        (
+            circuit.find_bit(bit).index,
+            tuple((reg[0].size, reg[0].name, reg[1]) for reg in circuit.find_bit(bit).registers),
+        )
+        for bit in bits
+    )
+
+
 def _circuit_key(circuit: QuantumCircuit, functional: bool = True) -> tuple:
     """Private key function for QuantumCircuit.
 
@@ -130,8 +141,14 @@ def _circuit_key(circuit: QuantumCircuit, functional: bool = True) -> tuple:
         circuit.num_qubits,
         circuit.num_clbits,
         circuit.num_parameters,
-        tuple(
-            (d.qubits, d.clbits, d.operation.name, tuple(d.operation.params)) for d in circuit.data
+        tuple(  # circuit.data
+            (
+                _bits_key(data.qubits, circuit),  # qubits
+                _bits_key(data.clbits, circuit),  # clbits
+                data.operation.name,  # operation.name
+                tuple(data.operation.params),  # operation.params
+            )
+            for data in circuit.data
         ),
         None if circuit._op_start_times is None else tuple(circuit._op_start_times),
     )