Add clifford gates to collect_cliffords (backport #12750) (#12763)

* Add clifford gates to collect_cliffords (#12750)

* add clifford gates to collect_cliffords

* replace hard coded clifford names by clifford_circuit names

* move import

* replace hard coded clifford names in random_clifford_circuit

* add release notes

* add permutation to collect_clifford gate list

(cherry picked from commit 41267ec)

# Conflicts:
#	qiskit/circuit/random/utils.py

* Update utils.py

* Update utils.py

---------

Co-authored-by: Shelly Garion <46566946+ShellyGarion@users.noreply.github.com>
Co-authored-by: Elena Peña Tapia <57907331+ElePT@users.noreply.github.com>

qiskit/transpiler/passes/optimization/collect_cliffords.py

@@ -22,6 +22,7 @@
 )
 
 from qiskit.quantum_info.operators import Clifford
+from qiskit.quantum_info.operators.symplectic.clifford_circuits import _BASIS_1Q, _BASIS_2Q
 
 
 class CollectCliffords(CollectAndCollapse):
@@ -69,21 +70,11 @@ def __init__(
         )
 
 
-clifford_gate_names = [
-    "x",
-    "y",
-    "z",
-    "h",
-    "s",
-    "sdg",
-    "cx",
-    "cy",
-    "cz",
-    "swap",
-    "clifford",
-    "linear_function",
-    "pauli",
-]
+clifford_gate_names = (
+    list(_BASIS_1Q.keys())
+    + list(_BASIS_2Q.keys())
+    + ["clifford", "linear_function", "pauli", "permutation"]
+)
 
 
 def _is_clifford_gate(node):

releasenotes/notes/fix-collect-clifford-83af26d98b8c69e8.yaml

@@ -0,0 +1,6 @@
+---
+fixes:
+  - |
+    Add more Clifford gates to the :class:`.CollectCliffords()` transpiler pass.
+    In particular, we have added the gates :class:`ECRGate()`, :class:`DCXGate()`,
+    :class:`iSWAPGate()`, :class:`SXGate()` and :class:`SXdgGate()` to this transpiler pass.