Transfer clifford and cnot-dihedral synthesis functions from quantum_…

…info to synthesis (#9141)

* transfer clifford synthesis functions from quantum_info to synthesis

* add init file

* transfer cnotdihedral synthesis from quantum_info to synthesis

* rename clifford synth functions. add docs

* add clifford synthesis methods to docs

* add documentation for cnotdihedral synth methods

* add decompose_cnotdihedral to quantum_info docs

* deprecate decompose_clifford and decompose_cnotdihedral - move to qsikit.synthesis

* style

* add clifford and cnotdihedral decompose full to synthesis

* enhance tests

* update high_level_synthesis transpiler pass

* add release notes

* fix docs in synthesis plugin

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

qiskit/quantum_info/operators/dihedral/dihedral.py

@@ -314,9 +314,9 @@ def to_circuit(self):
                *Scalable randomised benchmarking of non-Clifford gates*,
                npj Quantum Inf 2, 16012 (2016).
         """
-        from qiskit.quantum_info.synthesis.cnotdihedral_decompose import decompose_cnotdihedral
+        from qiskit.synthesis.cnotdihedral import synth_cnotdihedral_full
 
-        return decompose_cnotdihedral(self)
+        return synth_cnotdihedral_full(self)
 
     def to_instruction(self):
         """Return a Gate instruction implementing the CNOTDihedral object."""

qiskit/quantum_info/operators/symplectic/clifford.py

@@ -541,9 +541,9 @@ def to_circuit(self):
                Phys. Rev. A 70, 052328 (2004).
                `arXiv:quant-ph/0406196 <https://arxiv.org/abs/quant-ph/0406196>`_
         """
-        from qiskit.quantum_info.synthesis.clifford_decompose import decompose_clifford
+        from qiskit.synthesis.clifford import synth_clifford_full
 
-        return decompose_clifford(self)
+        return synth_clifford_full(self)
 
     def to_instruction(self):
         """Return a Gate instruction implementing the Clifford."""