Add representation of expr.Var to DAGCircuit (#12204)

* Add representation of `expr.Var` to `DAGCircuit`

This adds basic support for `Var`s to `DAGCircuit`, enabling the base
representation using the same wire structure used for clbits.  This is
known to be an inefficient representation of the dataflow for clbits,
which can be read multiple times without implying an order between those
operations (unlike qubits for which reads and writes are more naturally
linked).  We're using this simpler representation to make a better
initial MVP; optimising the data-flow representation would come as part
of a larger effort within the `DAGCircuit`.

This commit adds support in `DAGCircuit` for:

- representation of `Var` nodes
- appending all operations that might contain `Var` nodes to the DAG and
  updating the wire structure (including control-flow ops and stores)
- equality checking of DAGs with `Var`s (and so enables
  `QuantumCircuit.__eq__` as well)
- `DAGCircuit.copy_empty_like` with `Var`s
- the DAG/circuit converters

The other methods in `DAGCircuit` that might need to be aware of `Var`
nodes will be handled separately.

* Expand test coverage

* Fix copy/paste error

qiskit/circuit/controlflow/control_flow.py

@@ -22,6 +22,7 @@
 
 if typing.TYPE_CHECKING:
     from qiskit.circuit import QuantumCircuit
+    from qiskit.circuit.classical import expr
 
 
 class ControlFlowOp(Instruction, ABC):
@@ -72,3 +73,12 @@ def map_block(block: QuantumCircuit) -> QuantumCircuit:
         Returns:
             New :class:`ControlFlowOp` with replaced blocks.
         """
+
+    def iter_captured_vars(self) -> typing.Iterable[expr.Var]:
+        """Get an iterator over the unique captured variables in all blocks of this construct."""
+        seen = set()
+        for block in self.blocks:
+            for var in block.iter_captured_vars():
+                if var not in seen:
+                    seen.add(var)
+                    yield var

qiskit/converters/circuit_to_dag.py

@@ -79,6 +79,13 @@ def circuit_to_dag(circuit, copy_operations=True, *, qubit_order=None, clbit_ord
     dagcircuit.add_qubits(qubits)
     dagcircuit.add_clbits(clbits)
 
+    for var in circuit.iter_input_vars():
+        dagcircuit.add_input_var(var)
+    for var in circuit.iter_captured_vars():
+        dagcircuit.add_captured_var(var)
+    for var in circuit.iter_declared_vars():
+        dagcircuit.add_declared_var(var)
+
     for register in circuit.qregs:
         dagcircuit.add_qreg(register)
 

qiskit/converters/dag_to_circuit.py

@@ -62,7 +62,11 @@ def dag_to_circuit(dag, copy_operations=True):
         *dag.cregs.values(),
         name=name,
         global_phase=dag.global_phase,
+        inputs=dag.iter_input_vars(),
+        captures=dag.iter_captured_vars(),
     )
+    for var in dag.iter_declared_vars():
+        circuit.add_uninitialized_var(var)
     circuit.metadata = dag.metadata
     circuit.calibrations = dag.calibrations