diff --git a/qiskit/circuit/quantumcircuit.py b/qiskit/circuit/quantumcircuit.py
index 7c304f63af4a..a5182f47a11f 100644
--- a/qiskit/circuit/quantumcircuit.py
+++ b/qiskit/circuit/quantumcircuit.py
@@ -478,7 +478,9 @@ def __eq__(self, other) -> bool:
         # TODO: remove the DAG from this function
         from qiskit.converters import circuit_to_dag
 
-        return circuit_to_dag(self) == circuit_to_dag(other)
+        return circuit_to_dag(self, copy_operations=False) == circuit_to_dag(
+            other, copy_operations=False
+        )
 
     @classmethod
     def _increment_instances(cls):
diff --git a/qiskit/converters/circuit_to_dag.py b/qiskit/converters/circuit_to_dag.py
index 22cc81f1a89f..817f190fd2af 100644
--- a/qiskit/converters/circuit_to_dag.py
+++ b/qiskit/converters/circuit_to_dag.py
@@ -16,11 +16,18 @@
 from qiskit.dagcircuit.dagcircuit import DAGCircuit
 
 
-def circuit_to_dag(circuit):
+def circuit_to_dag(circuit, copy_operations=True):
     """Build a ``DAGCircuit`` object from a ``QuantumCircuit``.
 
     Args:
         circuit (QuantumCircuit): the input circuit.
+        copy_operations (bool): Deep copy the operation objects
+            in the :class:`~.QuantumCircuit` for the output :class:`~.DAGCircuit`.
+            This should only be set to ``False`` if the input :class:`~.QuantumCircuit`
+            will not be used anymore as the operations in the output
+            :class:`~.DAGCircuit` will be shared instances and modifications to
+            operations in the :class:`~.DAGCircuit` will be reflected in the
+            :class:`~.QuantumCircuit` (and vice versa).
 
     Return:
         DAGCircuit: the DAG representing the input circuit.
@@ -57,9 +64,10 @@ def circuit_to_dag(circuit):
         dagcircuit.add_creg(register)
 
     for instruction in circuit.data:
-        dagcircuit.apply_operation_back(
-            copy.deepcopy(instruction.operation), instruction.qubits, instruction.clbits
-        )
+        op = instruction.operation
+        if copy_operations:
+            op = copy.deepcopy(op)
+        dagcircuit.apply_operation_back(op, instruction.qubits, instruction.clbits)
 
     dagcircuit.duration = circuit.duration
     dagcircuit.unit = circuit.unit
diff --git a/releasenotes/notes/deepcopy-option-circuit_to_dag-1d494b7f9824ec93.yaml b/releasenotes/notes/deepcopy-option-circuit_to_dag-1d494b7f9824ec93.yaml
new file mode 100644
index 000000000000..043b37abe0bb
--- /dev/null
+++ b/releasenotes/notes/deepcopy-option-circuit_to_dag-1d494b7f9824ec93.yaml
@@ -0,0 +1,10 @@
+---
+features:
+  - |
+    Added a new option, ``copy_operations``, to :func:`~.circuit_to_dag` to
+    enable optionally disabling deep copying the operations from the input
+    :class`~.QuantumCircuit` to the output :class:`~.QuantumCircuit`. In cases
+    where the input :class`~.QuantumCircuit` is not used anymore after
+    conversion this deep copying is unnecessary overhead as any shared
+    references wouldn't have any potential unwanted side effects if the input
+    :class`~.QuantumCircuit` is discarded.