fix missing inverse definitions in generate_basis_approximations (Qiskit#13517)

* fix missing inverse definitions in `generate_basis_approximation`

* add missing basis gates to test

* add bugfix release note

* Update releasenotes/notes/fix-missing-inverse-definition-af7fe8d9f2193308.yaml

* Fix reno annotation

---------

Co-authored-by: Julien Gacon <[email protected]>
Co-authored-by: Julien Gacon <[email protected]>

Author: burgholzer

qiskit/synthesis/discrete_basis/generate_basis_approximations.py

@@ -37,6 +37,8 @@
     "tdg": "t",
     "s": "sdg",
     "sdg": "s",
+    "sx": "sxdg",
+    "sxdg": "sx",
 }
 
 _1q_gates = {

releasenotes/notes/fix-missing-inverse-definition-af7fe8d9f2193308.yaml

@@ -0,0 +1,6 @@
+fixes:
+  - |
+    Fixed a bug in the basis approximation generation for :class:`.SolovayKitaev`. 
+    Previously, generating discrete basis approximations using 
+    ``generate_basis_approximations`` for a basis containing ``"sx"`` or 
+    ``"sxdg"`` gates would fail. This has now been fixed.

test/python/transpiler/test_solovay_kitaev.py

@@ -463,7 +463,7 @@ def test_generate_basis_approximation_gates(self):
 
         Regression test of Qiskit/qiskit-terra#9585.
         """
-        basis = ["i", "x", "y", "z", "h", "t", "tdg", "s", "sdg"]
+        basis = ["i", "x", "y", "z", "h", "t", "tdg", "s", "sdg", "sx", "sxdg"]
         approx = generate_basic_approximations(basis, depth=2)
 
         # This mainly checks that there are no errors in the generation (like