diff --git a/crates/circuit/src/circuit_data.rs b/crates/circuit/src/circuit_data.rs
index 07f4579a4cd3..58e657a574de 100644
--- a/crates/circuit/src/circuit_data.rs
+++ b/crates/circuit/src/circuit_data.rs
@@ -18,7 +18,7 @@ use crate::circuit_instruction::{
     convert_py_to_operation_type, CircuitInstruction, ExtraInstructionAttributes, OperationInput,
     PackedInstruction,
 };
-use crate::imports::{BUILTIN_LIST, QUBIT};
+use crate::imports::{BUILTIN_LIST, DEEPCOPY, QUBIT};
 use crate::interner::{IndexedInterner, Interner, InternerKey};
 use crate::operations::{Operation, OperationType, Param, StandardGate};
 use crate::parameter_table::{ParamEntry, ParamTable, GLOBAL_PHASE_INDEX};
@@ -487,20 +487,17 @@ impl CircuitData {
         res.param_table.clone_from(&self.param_table);
 
         if deepcopy {
-            let deepcopy = py
-                .import_bound(intern!(py, "copy"))?
-                .getattr(intern!(py, "deepcopy"))?;
             for inst in &mut res.data {
                 match &mut inst.op {
                     OperationType::Standard(_) => {}
                     OperationType::Gate(ref mut op) => {
-                        op.gate = deepcopy.call1((&op.gate,))?.unbind();
+                        op.gate = DEEPCOPY.get_bound(py).call1((&op.gate,))?.unbind();
                     }
                     OperationType::Instruction(ref mut op) => {
-                        op.instruction = deepcopy.call1((&op.instruction,))?.unbind();
+                        op.instruction = DEEPCOPY.get_bound(py).call1((&op.instruction,))?.unbind();
                     }
                     OperationType::Operation(ref mut op) => {
-                        op.operation = deepcopy.call1((&op.operation,))?.unbind();
+                        op.operation = DEEPCOPY.get_bound(py).call1((&op.operation,))?.unbind();
                     }
                 };
                 #[cfg(feature = "cache_pygates")]
diff --git a/crates/circuit/src/imports.rs b/crates/circuit/src/imports.rs
index 530e635c94f1..e2a1ca542453 100644
--- a/crates/circuit/src/imports.rs
+++ b/crates/circuit/src/imports.rs
@@ -71,6 +71,7 @@ pub static SINGLETON_GATE: ImportOnceCell =
     ImportOnceCell::new("qiskit.circuit.singleton", "SingletonGate");
 pub static SINGLETON_CONTROLLED_GATE: ImportOnceCell =
     ImportOnceCell::new("qiskit.circuit.singleton", "SingletonControlledGate");
+pub static DEEPCOPY: ImportOnceCell = ImportOnceCell::new("copy", "deepcopy");
 
 pub static WARNINGS_WARN: ImportOnceCell = ImportOnceCell::new("warnings", "warn");
 
diff --git a/crates/circuit/src/operations.rs b/crates/circuit/src/operations.rs
index 1cd7f3702b50..3fe808876a70 100644
--- a/crates/circuit/src/operations.rs
+++ b/crates/circuit/src/operations.rs
@@ -13,7 +13,7 @@
 use std::f64::consts::PI;
 
 use crate::circuit_data::CircuitData;
-use crate::imports::{PARAMETER_EXPRESSION, QUANTUM_CIRCUIT};
+use crate::imports::{DEEPCOPY, PARAMETER_EXPRESSION, QUANTUM_CIRCUIT};
 use crate::{gate_matrix, Qubit};
 
 use ndarray::{aview2, Array2};
@@ -1185,6 +1185,16 @@ impl PyInstruction {
             instruction,
         }
     }
+
+    fn __deepcopy__(&self, py: Python, _memo: PyObject) -> PyResult<Self> {
+        Ok(PyInstruction {
+            qubits: self.qubits,
+            clbits: self.clbits,
+            params: self.params,
+            op_name: self.op_name.clone(),
+            instruction: DEEPCOPY.get_bound(py).call1((&self.instruction,))?.unbind(),
+        })
+    }
 }
 
 impl Operation for PyInstruction {
@@ -1264,6 +1274,16 @@ impl PyGate {
             gate,
         }
     }
+
+    fn __deepcopy__(&self, py: Python, _memo: PyObject) -> PyResult<Self> {
+        Ok(PyGate {
+            qubits: self.qubits,
+            clbits: self.clbits,
+            params: self.params,
+            op_name: self.op_name.clone(),
+            gate: DEEPCOPY.get_bound(py).call1((&self.gate,))?.unbind(),
+        })
+    }
 }
 
 impl Operation for PyGate {
@@ -1356,6 +1376,16 @@ impl PyOperation {
             operation,
         }
     }
+
+    fn __deepcopy__(&self, py: Python, _memo: PyObject) -> PyResult<Self> {
+        Ok(PyOperation {
+            qubits: self.qubits,
+            clbits: self.clbits,
+            params: self.params,
+            op_name: self.op_name.clone(),
+            operation: DEEPCOPY.get_bound(py).call1((&self.operation,))?.unbind(),
+        })
+    }
 }
 
 impl Operation for PyOperation {
diff --git a/qiskit/converters/circuit_to_dag.py b/qiskit/converters/circuit_to_dag.py
index b2c1df2a037b..88d9c72f1d61 100644
--- a/qiskit/converters/circuit_to_dag.py
+++ b/qiskit/converters/circuit_to_dag.py
@@ -13,7 +13,8 @@
 """Helper function for converting a circuit to a dag"""
 import copy
 
-from qiskit.dagcircuit.dagcircuit import DAGCircuit
+from qiskit.dagcircuit.dagcircuit import DAGCircuit, DAGOpNode
+from qiskit._accelerate.circuit import StandardGate
 
 
 def circuit_to_dag(circuit, copy_operations=True, *, qubit_order=None, clbit_order=None):
@@ -93,10 +94,24 @@ def circuit_to_dag(circuit, copy_operations=True, *, qubit_order=None, clbit_ord
         dagcircuit.add_creg(register)
 
     for instruction in circuit.data:
-        op = instruction.operation
-        if copy_operations:
-            op = copy.deepcopy(op)
-        dagcircuit.apply_operation_back(op, instruction.qubits, instruction.clbits, check=False)
+        if not isinstance(instruction._raw_op, StandardGate):
+            op = instruction.operation
+            if copy_operations:
+                op = copy.deepcopy(op)
+            dagcircuit.apply_operation_back(op, instruction.qubits, instruction.clbits, check=False)
+        else:
+            node = DAGOpNode(
+                instruction._raw_op,
+                qargs=instruction.qubits,
+                cargs=instruction.clbits,
+                params=instruction.params,
+                label=instruction.label,
+                duration=instruction.duration,
+                unit=instruction.unit,
+                condition=instruction.condition,
+                dag=dagcircuit,
+            )
+            dagcircuit._apply_op_node_back(node)
 
     dagcircuit.duration = circuit.duration
     dagcircuit.unit = circuit.unit
diff --git a/qiskit/dagcircuit/dagcircuit.py b/qiskit/dagcircuit/dagcircuit.py
index a0ea2aec6420..0213d242097f 100644
--- a/qiskit/dagcircuit/dagcircuit.py
+++ b/qiskit/dagcircuit/dagcircuit.py
@@ -718,6 +718,11 @@ def copy_empty_like(self, *, vars_mode: _VarsMode = "alike"):
         return target_dag
 
     def _apply_op_node_back(self, node: DAGOpNode):
+        additional = ()
+        if _may_have_additional_wires(node):
+            # This is the slow path; most of the time, this won't happen.
+            additional = set(_additional_wires(node)).difference(node.cargs)
+
         node._node_id = self._multi_graph.add_node(node)
         self._increment_op(node)
 
@@ -726,7 +731,11 @@ def _apply_op_node_back(self, node: DAGOpNode):
         # and adding new edges from the operation node to each output node
         self._multi_graph.insert_node_on_in_edges_multiple(
             node._node_id,
-            [self.output_map[bit]._node_id for bits in (node.qargs, node.cargs) for bit in bits],
+            [
+                self.output_map[bit]._node_id
+                for bits in (node.qargs, node.cargs, additional)
+                for bit in bits
+            ],
         )
 
     def apply_operation_back(