diff --git a/qiskit/circuit/library/standard_gates/equivalence_library.py b/qiskit/circuit/library/standard_gates/equivalence_library.py
index 9793dc0202a8..c4619ca27858 100644
--- a/qiskit/circuit/library/standard_gates/equivalence_library.py
+++ b/qiskit/circuit/library/standard_gates/equivalence_library.py
@@ -850,6 +850,56 @@ def _cnot_rxx_decompose(plus_ry: bool = True, plus_rxx: bool = True):
     def_swap.append(inst, qargs, cargs)
 _sel.add_equivalence(SwapGate(), def_swap)
 
+# SwapGate
+#
+# q_0: ─X─
+#       │   ≡
+# q_1: ─X─
+#
+#      ┌──────────┐┌──────┐   ┌────┐   ┌──────┐┌──────────┐┌──────┐
+# q_0: ┤ Rz(-π/2) ├┤0     ├───┤ √X ├───┤1     ├┤ Rz(-π/2) ├┤0     ├
+#      └──┬────┬──┘│  Ecr │┌──┴────┴──┐│  Ecr │└──┬────┬──┘│  Ecr │
+# q_1: ───┤ √X ├───┤1     ├┤ Rz(-π/2) ├┤0     ├───┤ √X ├───┤1     ├
+#         └────┘   └──────┘└──────────┘└──────┘   └────┘   └──────┘
+#
+q = QuantumRegister(2, "q")
+def_swap_ecr = QuantumCircuit(q)
+def_swap_ecr.rz(-pi / 2, 0)
+def_swap_ecr.sx(1)
+def_swap_ecr.ecr(0, 1)
+def_swap_ecr.rz(-pi / 2, 1)
+def_swap_ecr.sx(0)
+def_swap_ecr.ecr(1, 0)
+def_swap_ecr.rz(-pi / 2, 0)
+def_swap_ecr.sx(1)
+def_swap_ecr.ecr(0, 1)
+_sel.add_equivalence(SwapGate(), def_swap_ecr)
+
+# SwapGate
+#
+# q_0: ─X─
+#       │   ≡
+# q_1: ─X─
+#
+# global phase: 3π/2
+#      ┌────┐   ┌────┐   ┌────┐
+# q_0: ┤ √X ├─■─┤ √X ├─■─┤ √X ├─■─
+#      ├────┤ │ ├────┤ │ ├────┤ │
+# q_1: ┤ √X ├─■─┤ √X ├─■─┤ √X ├─■─
+#      └────┘   └────┘   └────┘
+q = QuantumRegister(2, "q")
+def_swap_cz = QuantumCircuit(q, global_phase=-pi / 2)
+def_swap_cz.sx(0)
+def_swap_cz.sx(1)
+def_swap_cz.cz(0, 1)
+def_swap_cz.sx(0)
+def_swap_cz.sx(1)
+def_swap_cz.cz(0, 1)
+def_swap_cz.sx(0)
+def_swap_cz.sx(1)
+def_swap_cz.cz(0, 1)
+_sel.add_equivalence(SwapGate(), def_swap_cz)
+
 # iSwapGate
 #
 #      ┌────────┐          ┌───┐┌───┐     ┌───┐
diff --git a/test/python/circuit/test_gate_definitions.py b/test/python/circuit/test_gate_definitions.py
index 8e608a163900..38bf7046cae5 100644
--- a/test/python/circuit/test_gate_definitions.py
+++ b/test/python/circuit/test_gate_definitions.py
@@ -21,6 +21,7 @@
 from qiskit import QuantumCircuit, QuantumRegister
 from qiskit.quantum_info import Operator
 from qiskit.circuit import ParameterVector, Gate, ControlledGate
+from qiskit.circuit.singleton import SingletonGate, SingletonControlledGate
 from qiskit.circuit.library import standard_gates
 from qiskit.circuit.library import (
     HGate,
@@ -260,7 +261,12 @@ class TestGateEquivalenceEqual(QiskitTestCase):
     """Test the decomposition of a gate in terms of other gates
     yields the same matrix as the hardcoded matrix definition."""
 
-    class_list = Gate.__subclasses__() + ControlledGate.__subclasses__()
+    class_list = (
+        SingletonGate.__subclasses__()
+        + SingletonControlledGate.__subclasses__()
+        + Gate.__subclasses__()
+        + ControlledGate.__subclasses__()
+    )
     exclude = {
         "ControlledGate",
         "DiagonalGate",
@@ -313,7 +319,11 @@ def test_equivalence_phase(self, gate_class):
             with self.subTest(msg=gate.name + "_" + str(ieq)):
                 op1 = Operator(gate)
                 op2 = Operator(equivalency)
-                self.assertEqual(op1, op2)
+                msg = (
+                    f"Equivalence entry from '{gate.name}' to:\n"
+                    f"{str(equivalency.draw('text'))}\nfailed"
+                )
+                self.assertEqual(op1, op2, msg)
 
 
 @ddt
diff --git a/test/qpy_compat/test_qpy.py b/test/qpy_compat/test_qpy.py
index 58ee1abc2a2f..438b002f11bd 100755
--- a/test/qpy_compat/test_qpy.py
+++ b/test/qpy_compat/test_qpy.py
@@ -661,7 +661,7 @@ def generate_annotated_circuits():
         CXGate(), [InverseModifier(), ControlModifier(1), PowerModifier(1.4), InverseModifier()]
     )
     op2 = AnnotatedOperation(XGate(), InverseModifier())
-    qc = QuantumCircuit(6, 1)
+    qc = QuantumCircuit(6, 1, name="Annotated circuits")
     qc.cx(0, 1)
     qc.append(op1, [0, 1, 2])
     qc.h(4)