Optimization for the MCX Recursive Gate

Change the recursive method for the Lemma 9 of arXiv:1501.06911, first shown in Lemma 7.3 of https://link.aps.org/doi/10.1103/PhysRevA.52.3457

Co-Authored-By: Rafaella Vale <26910380+rafaella-vale@users.noreply.github.com>
Co-Authored-By: Jefferson Deyvis <67497412+jeffersondeyvis@users.noreply.github.com>

qiskit/circuit/library/standard_gates/x.py

@@ -1093,8 +1093,22 @@ def _define(self):
             qc._append(C4XGate(), q[:], [])
             self.definition = qc
         else:
-            for instr, qargs, cargs in self._recurse(q[:-1], q_ancilla=q[-1]):
-                qc._append(instr, qargs, cargs)
+            num_ctrl_qubits = len(q) - 1
+            q_ancilla = q[-1]
+            q_target = q[-2]
+            middle = ceil(num_ctrl_qubits / 2)
+            first_half = [*q[:middle]]
+            second_half = [*q[middle:num_ctrl_qubits - 1], q_ancilla]
+
+            qc._append(MCXVChain(num_ctrl_qubits=len(first_half), dirty_ancillas=True), \
+                        [*first_half, q_ancilla, *q[middle:middle + len(first_half) - 2]], [])
+            qc._append(MCXVChain(num_ctrl_qubits=len(second_half), dirty_ancillas=True), \
+                        [*second_half, q_target, *q[:len(second_half) - 2]], [])
+            qc._append(MCXVChain(num_ctrl_qubits=len(first_half), dirty_ancillas=True), \
+                        [*first_half, q_ancilla, *q[middle:middle + len(first_half) - 2]], [])
+            qc._append(MCXVChain(num_ctrl_qubits=len(second_half), dirty_ancillas=True), \
+                        [*second_half, q_target, *q[:len(second_half) - 2]], [])
+
             self.definition = qc
 
     def _recurse(self, q, q_ancilla=None):