Fix inverse of SdgGate in Solovay Kitaev (#9635)

* Fix typo and add regression test

* add example and include basic_approx in API doc

* use assertTrue

* use `assertGreater`

* run black with the right version this time

---------

Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>

qiskit/synthesis/__init__.py

@@ -77,6 +77,7 @@
    :toctree: ../stubs/
 
    SolovayKitaevDecomposition
+   generate_basic_approximations
 
 """
 
@@ -107,4 +108,4 @@
     synth_cnotdihedral_two_qubits,
     synth_cnotdihedral_general,
 )
-from .discrete_basis import SolovayKitaevDecomposition
+from .discrete_basis import SolovayKitaevDecomposition, generate_basic_approximations

qiskit/synthesis/discrete_basis/__init__.py

@@ -13,3 +13,4 @@
 """Discrete basis synthesis algorithms."""
 
 from .solovay_kitaev import SolovayKitaevDecomposition
+from .generate_basis_approximations import generate_basic_approximations

qiskit/synthesis/discrete_basis/generate_basis_approximations.py

@@ -36,7 +36,7 @@
     "t": "tdg",
     "tdg": "t",
     "s": "sdg",
-    "sgd": "s",
+    "sdg": "s",
 }
 
 _1q_gates = {

qiskit/transpiler/passes/synthesis/solovay_kitaev_synthesis.py

@@ -63,11 +63,12 @@ class SolovayKitaev(TransformationPass):
 
     Examples:
 
+        Per default, the basis gate set is ``["t", "tdg", "h"]``:
+
         .. code-block::
 
             import numpy as np
             from qiskit.circuit import QuantumCircuit
-            from qiskit.circuit.library import TGate, HGate, TdgGate
             from qiskit.transpiler.passes.synthesis import SolovayKitaev
             from qiskit.quantum_info import Operator
 
@@ -77,7 +78,6 @@ class SolovayKitaev(TransformationPass):
             print("Original circuit:")
             print(circuit.draw())
 
-            basis_gates = [TGate(), TdgGate(), HGate()]
             skd = SolovayKitaev(recursion_degree=2)
 
             discretized = skd(circuit)
@@ -100,6 +100,18 @@ class SolovayKitaev(TransformationPass):
                └───┘└───┘└───┘
             Error: 2.828408279166474
 
+        For individual basis gate sets, the ``generate_basic_approximations`` function can be used:
+
+        .. code-block::
+
+            from qiskit.synthesis import generate_basic_approximations
+            from qiskit.transpiler.passes import SolovayKitaev
+
+            basis = ["s", "sdg", "t", "tdg", "z", "h"]
+            approx = generate_basic_approximations(basis, depth=3)
+
+            skd = SolovayKitaev(recursion_degree=2, basic_approximations=approx)
+
     References:
 
         [1]: Kitaev, A Yu (1997). Quantum computations: algorithms and error correction.

releasenotes/notes/fix-sk-sdg-81ec87abe7af4a89.yaml

@@ -0,0 +1,6 @@
+---
+fixes:
+  - |
+    Fixed a bug in :func:`.generate_basic_approximations` where the inverse of the 
+    :class:`.SdgGate` was not correctly recognized as :class:`.SGate`.
+    Fixed `#9585 <https://github.com/Qiskit/qiskit-terra/issues/9585>`__.

test/python/transpiler/test_solovay_kitaev.py

@@ -394,6 +394,18 @@ def test_commutator_decompose_decomposes_correctly(self, u_so3):
         actual_commutator = np.dot(v_so3, np.dot(w_so3, np.dot(np.conj(v_so3).T, np.conj(w_so3).T)))
         self.assertTrue(np.allclose(actual_commutator, u_so3))
 
+    def test_generate_basis_approximation_gates(self):
+        """Test the basis approximation generation works for all supported gates.
+
+        Regression test of Qiskit/qiskit-terra#9585.
+        """
+        basis = ["i", "x", "y", "z", "h", "t", "tdg", "s", "sdg"]
+        approx = generate_basic_approximations(basis, depth=2)
+
+        # This mainly checks that there are no errors in the generation (like
+        # in computing the inverse as described in #9585), so a simple check is enough.
+        self.assertGreater(len(approx), len(basis))
+
 
 if __name__ == "__main__":
     unittest.main()