Update CS and CSdg in the equivalence library (Qiskit#14001)

* update CS and CSdg in the equivalence library

* add release notes

* update release notes following review

* fix misprint

* add back original equivalences

* update releasde notes

* Update releasenotes/notes/update-cs-csdg-in-equivalence-library-c1e70d3246f4aa6d.yaml

Co-authored-by: Alexander Ivrii <[email protected]>

---------

Co-authored-by: Alexander Ivrii <[email protected]>

Author: ShellyGarion

qiskit/circuit/library/standard_gates/equivalence_library.py

@@ -807,33 +807,6 @@ def _cnot_rxx_decompose(plus_ry: bool = True, plus_rxx: bool = True):
 def_sdg.append(U1Gate(-pi / 2), [q[0]], [])
 _sel.add_equivalence(SdgGate(), def_sdg)
 
-# CSGate
-#
-# q_0: ──■──   q_0: ───────■────────
-#      ┌─┴─┐        ┌───┐┌─┴──┐┌───┐
-# q_1: ┤ S ├ = q_1: ┤ H ├┤ Sx ├┤ H ├
-#      └───┘        └───┘└────┘└───┘
-q = QuantumRegister(2, "q")
-def_cs = QuantumCircuit(q)
-def_cs.append(HGate(), [q[1]], [])
-def_cs.append(CSXGate(), [q[0], q[1]], [])
-def_cs.append(HGate(), [q[1]], [])
-_sel.add_equivalence(CSGate(), def_cs)
-
-# CSdgGate
-#
-# q_0: ───■───   q_0: ───────■────■────────
-#      ┌──┴──┐        ┌───┐┌─┴─┐┌─┴──┐┌───┐
-# q_1: ┤ Sdg ├ = q_1: ┤ H ├┤ X ├┤ Sx ├┤ H ├
-#      └─────┘        └───┘└───┘└────┘└───┘
-q = QuantumRegister(2, "q")
-def_csdg = QuantumCircuit(q)
-def_csdg.append(HGate(), [q[1]], [])
-def_csdg.append(CXGate(), [q[0], q[1]], [])
-def_csdg.append(CSXGate(), [q[0], q[1]], [])
-def_csdg.append(HGate(), [q[1]], [])
-_sel.add_equivalence(CSdgGate(), def_csdg)
-
 # SdgGate
 #
 #    ┌─────┐        ┌───┐┌───┐
@@ -877,6 +850,65 @@ def _cnot_rxx_decompose(plus_ry: bool = True, plus_rxx: bool = True):
     def_sdg.append(inst, qargs, cargs)
 _sel.add_equivalence(SdgGate(), def_sdg)
 
+# CSGate
+#
+#                    ┌───┐
+# q_0: ──■──    q_0: ┤ T ├──■───────────■──
+#      ┌─┴─┐         ├───┤┌─┴─┐┌─────┐┌─┴─┐
+# q_1: ┤ S ├ =  q_1: ┤ T ├┤ X ├┤ Tdg ├┤ X ├
+#      └───┘         └───┘└───┘└─────┘└───┘
+q = QuantumRegister(2, "q")
+def_cs = QuantumCircuit(q)
+def_cs.append(TGate(), [q[0]], [])
+def_cs.append(TGate(), [q[1]], [])
+def_cs.append(CXGate(), [q[0], q[1]], [])
+def_cs.append(TdgGate(), [q[1]], [])
+def_cs.append(CXGate(), [q[0], q[1]], [])
+_sel.add_equivalence(CSGate(), def_cs)
+
+# CSGate
+#
+# q_0: ──■──   q_0: ───────■────────
+#      ┌─┴─┐        ┌───┐┌─┴──┐┌───┐
+# q_1: ┤ S ├ = q_1: ┤ H ├┤ Sx ├┤ H ├
+#      └───┘        └───┘└────┘└───┘
+q = QuantumRegister(2, "q")
+def_cs_csx = QuantumCircuit(q)
+def_cs_csx.append(HGate(), [q[1]], [])
+def_cs_csx.append(CSXGate(), [q[0], q[1]], [])
+def_cs_csx.append(HGate(), [q[1]], [])
+_sel.add_equivalence(CSGate(), def_cs_csx)
+
+# CSdgGate
+#
+#                                     ┌─────┐
+# q_0: ───■───    q_0: ──■─────────■──┤ Tdg ├
+#      ┌──┴──┐         ┌─┴─┐┌───┐┌─┴─┐├─────┤
+# q_1: ┤ Sdg ├ =  q_1: ┤ X ├┤ T ├┤ X ├┤ Tdg ├
+#      └─────┘         └───┘└───┘└───┘└─────┘
+q = QuantumRegister(2, "q")
+def_csdg = QuantumCircuit(q)
+def_csdg.append(CXGate(), [q[0], q[1]], [])
+def_csdg.append(TGate(), [q[1]], [])
+def_csdg.append(CXGate(), [q[0], q[1]], [])
+def_csdg.append(TdgGate(), [q[0]], [])
+def_csdg.append(TdgGate(), [q[1]], [])
+_sel.add_equivalence(CSdgGate(), def_csdg)
+
+# CSdgGate
+#
+# q_0: ───■───   q_0: ───────■────■────────
+#      ┌──┴──┐        ┌───┐┌─┴─┐┌─┴──┐┌───┐
+# q_1: ┤ Sdg ├ = q_1: ┤ H ├┤ X ├┤ Sx ├┤ H ├
+#      └─────┘        └───┘└───┘└────┘└───┘
+q = QuantumRegister(2, "q")
+def_csdg_csx = QuantumCircuit(q)
+def_csdg_csx.append(HGate(), [q[1]], [])
+def_csdg_csx.append(CXGate(), [q[0], q[1]], [])
+def_csdg_csx.append(CSXGate(), [q[0], q[1]], [])
+def_csdg_csx.append(HGate(), [q[1]], [])
+_sel.add_equivalence(CSdgGate(), def_csdg_csx)
+
 # SwapGate
 #                        ┌───┐
 # q_0: ─X─     q_0: ──■──┤ X ├──■──

releasenotes/notes/update-cs-csdg-in-equivalence-library-c1e70d3246f4aa6d.yaml

@@ -0,0 +1,7 @@
+---
+features_synthesis:
+  - |
+    The standard equivalence library has additional equivalences for :class:`.CSGate` and :class:`.CSdgGate`.
+    When transpiling to the `["cx", "u"]` basis set, the :class:`.BasisTranslator` transpiler pass now uses
+    2  :class:`.CXGate` gates instead of 3 for :class:`.CSdgGate`, and in addition reduces the number of single-qubit
+    gates in both cases.