From 507d5b3a89363634ef51779595d00f618ca58f88 Mon Sep 17 00:00:00 2001
From: Thiago Melo <thiagomdazevedo@hotmail.com>
Date: Wed, 7 Sep 2022 17:30:40 -0300
Subject: [PATCH 01/17] 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 | 18 ++++++++++++++++--
 1 file changed, 16 insertions(+), 2 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index cc6eebc2f96e..42d0afc552f9 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/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):

From 4671b7e32bf9e689b35f07e3464ceffddb99fa30 Mon Sep 17 00:00:00 2001
From: Thiago Melo <thiagomdazevedo@hotmail.com>
Date: Wed, 7 Sep 2022 17:39:59 -0300
Subject: [PATCH 02/17] Revert "Optimization for the MCX Recursive Gate"

This reverts commit 507d5b3a89363634ef51779595d00f618ca58f88.

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 | 18 ++----------------
 1 file changed, 2 insertions(+), 16 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index 42d0afc552f9..cc6eebc2f96e 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/qiskit/circuit/library/standard_gates/x.py
@@ -1093,22 +1093,8 @@ def _define(self):
             qc._append(C4XGate(), q[:], [])
             self.definition = qc
         else:
-            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]], [])
-
+            for instr, qargs, cargs in self._recurse(q[:-1], q_ancilla=q[-1]):
+                qc._append(instr, qargs, cargs)
             self.definition = qc
 
     def _recurse(self, q, q_ancilla=None):

From af4c3f68aaa61ba1f405c382546f2cc97cd119d3 Mon Sep 17 00:00:00 2001
From: Thiago Melo <thiagomdazevedo@hotmail.com>
Date: Wed, 7 Sep 2022 17:43:39 -0300
Subject: [PATCH 03/17] Revert "Revert "Optimization for the MCX Recursive
 Gate""

This reverts commit 4671b7e32bf9e689b35f07e3464ceffddb99fa30.
---
 qiskit/circuit/library/standard_gates/x.py | 18 ++++++++++++++++--
 1 file changed, 16 insertions(+), 2 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index cc6eebc2f96e..42d0afc552f9 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/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):

From 8ca0031b1af9c107390eac1e6f00b9f86865f37f Mon Sep 17 00:00:00 2001
From: Thiago Melo <thiagomdazevedo@hotmail.com>
Date: Wed, 7 Sep 2022 17:45:40 -0300
Subject: [PATCH 04/17] Optimization for MCX Recursive

Fixing co-authors

Co-Authored-By: Rafaella Vale <26910380+rafaella-vale@users.noreply.github.com>
Co-Authored-By: Jefferson Deyvis <67497412+jeffersondeyvis@users.noreply.github.com>
Co-Authored-By: Adenilton Silva <7927558+adjs@users.noreply.github.com>
---
 qiskit/circuit/library/standard_gates/x.py | 18 ++----------------
 1 file changed, 2 insertions(+), 16 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index 42d0afc552f9..cc6eebc2f96e 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/qiskit/circuit/library/standard_gates/x.py
@@ -1093,22 +1093,8 @@ def _define(self):
             qc._append(C4XGate(), q[:], [])
             self.definition = qc
         else:
-            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]], [])
-
+            for instr, qargs, cargs in self._recurse(q[:-1], q_ancilla=q[-1]):
+                qc._append(instr, qargs, cargs)
             self.definition = qc
 
     def _recurse(self, q, q_ancilla=None):

From e3729c9075efc5cb6b00561b19580d1fc31c8260 Mon Sep 17 00:00:00 2001
From: Thiago Melo <thiagomdazevedo@hotmail.com>
Date: Wed, 7 Sep 2022 17:46:33 -0300
Subject: [PATCH 05/17] Optimization of MCX Recursive

Now with fixed co-authors

Co-Authored-By: Rafaella Vale <26910380+rafaella-vale@users.noreply.github.com>
Co-Authored-By: Jefferson Deyvis <67497412+jeffersondeyvis@users.noreply.github.com>
Co-Authored-By: Adenilton Silva <7927558+adjs@users.noreply.github.com>
---
 qiskit/circuit/library/standard_gates/x.py | 18 ++++++++++++++++--
 1 file changed, 16 insertions(+), 2 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index cc6eebc2f96e..42d0afc552f9 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/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):

From 4f10325c6468aaba859c229d8ff8172e628eacd4 Mon Sep 17 00:00:00 2001
From: rafaella-vale <26910380+rafaella-vale@users.noreply.github.com>
Date: Thu, 8 Sep 2022 11:05:31 -0300
Subject: [PATCH 06/17] refactored MCXRecursive and unused method deleted

---
 qiskit/circuit/library/standard_gates/x.py | 53 +++++++++-------------
 1 file changed, 21 insertions(+), 32 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index 42d0afc552f9..a99cd3159ea4 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/qiskit/circuit/library/standard_gates/x.py
@@ -1098,42 +1098,31 @@ def _define(self):
             q_target = q[-2]
             middle = ceil(num_ctrl_qubits / 2)
             first_half = [*q[:middle]]
-            second_half = [*q[middle:num_ctrl_qubits - 1], q_ancilla]
+            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]], [])
+            qc._append(
+                MCXVChain(num_ctrl_qubits=len(first_half), dirty_ancillas=True),
+                qargs=[*first_half, q_ancilla, *q[middle : middle + len(first_half) - 2]],
+                cargs=[],
+            )
+            qc._append(
+                MCXVChain(num_ctrl_qubits=len(second_half), dirty_ancillas=True),
+                qargs=[*second_half, q_target, *q[: len(second_half) - 2]],
+                cargs=[],
+            )
+            qc._append(
+                MCXVChain(num_ctrl_qubits=len(first_half), dirty_ancillas=True),
+                qargs=[*first_half, q_ancilla, *q[middle : middle + len(first_half) - 2]],
+                cargs=[],
+            )
+            qc._append(
+                MCXVChain(num_ctrl_qubits=len(second_half), dirty_ancillas=True),
+                qargs=[*second_half, q_target, *q[: len(second_half) - 2]],
+                cargs=[],
+            )
 
             self.definition = qc
 
-    def _recurse(self, q, q_ancilla=None):
-        # recursion stop
-        if len(q) == 4:
-            return [(C3XGate(), q[:], [])]
-        if len(q) == 5:
-            return [(C4XGate(), q[:], [])]
-        if len(q) < 4:
-            raise AttributeError("Something went wrong in the recursion, have less than 4 qubits.")
-
-        # recurse
-        num_ctrl_qubits = len(q) - 1
-        middle = ceil(num_ctrl_qubits / 2)
-        first_half = [*q[:middle], q_ancilla]
-        second_half = [*q[middle:num_ctrl_qubits], q_ancilla, q[num_ctrl_qubits]]
-
-        rule = []
-        rule += self._recurse(first_half, q_ancilla=q[middle])
-        rule += self._recurse(second_half, q_ancilla=q[middle - 1])
-        rule += self._recurse(first_half, q_ancilla=q[middle])
-        rule += self._recurse(second_half, q_ancilla=q[middle - 1])
-
-        return rule
-
 
 class MCXVChain(MCXGate):
     """Implement the multi-controlled X gate using a V-chain of CX gates."""

From a891dc4e34232e66f352dbf51ed66fa2585f9bc3 Mon Sep 17 00:00:00 2001
From: rafaella-vale <26910380+rafaella-vale@users.noreply.github.com>
Date: Thu, 8 Sep 2022 17:42:31 -0300
Subject: [PATCH 07/17] fixed qasm string for mcx test with variants

---
 test/python/circuit/test_circuit_qasm.py | 17 +++++++++++------
 1 file changed, 11 insertions(+), 6 deletions(-)

diff --git a/test/python/circuit/test_circuit_qasm.py b/test/python/circuit/test_circuit_qasm.py
index cd3b4fe90fac..452a45abd12d 100644
--- a/test/python/circuit/test_circuit_qasm.py
+++ b/test/python/circuit/test_circuit_qasm.py
@@ -270,6 +270,8 @@ def test_circuit_qasm_with_mcx_gate(self):
         qc = QuantumCircuit(4)
         qc.mcx([0, 1, 2], 3)
 
+        print(qc.decompose().draw())
+
         # qasm output doesn't support parameterized gate yet.
         # param0 for "gate mcuq(param0) is not used inside the definition
         expected_qasm = """OPENQASM 2.0;
@@ -288,20 +290,23 @@ def test_circuit_qasm_with_mcx_gate_variants(self):
         qc.append(cl.MCXGrayCode(n), range(n + 1))
         qc.append(cl.MCXRecursive(n), range(n + 2))
         qc.append(cl.MCXVChain(n), range(2 * n - 1))
+        mcx_vchain_id = id(qc.data[-1].operation)
 
         # qasm output doesn't support parameterized gate yet.
         # param0 for "gate mcuq(param0) is not used inside the definition
         expected_qasm = """OPENQASM 2.0;
 include "qelib1.inc";
-gate mcx q0,q1,q2,q3 { h q3; p(pi/8) q0; p(pi/8) q1; p(pi/8) q2; p(pi/8) q3; cx q0,q1; p(-pi/8) q1; cx q0,q1; cx q1,q2; p(-pi/8) q2; cx q0,q2; p(pi/8) q2; cx q1,q2; p(-pi/8) q2; cx q0,q2; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; h q3; }
-gate mcu1(param0) q0,q1,q2,q3,q4,q5 { cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }
-gate mcx_gray q0,q1,q2,q3,q4,q5 { h q5; mcu1(pi) q0,q1,q2,q3,q4,q5; h q5; }
-gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 { mcx q0,q1,q2,q6; mcx q3,q4,q6,q5; mcx q0,q1,q2,q6; mcx q3,q4,q6,q5; }
-gate mcx_vchain q0,q1,q2,q3,q4,q5,q6,q7,q8 { rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }
+gate mcx_vchain q0,q1,q2,q3,q4 {{ u2(0,pi) q3; cx q3,q4; u1(-pi/4) q4; cx q2,q4; u1(pi/4) q4; cx q3,q4; u1(-pi/4) q4; cx q2,q4; u1(pi/4) q4; rccx q0,q1,q4; u1(-pi/4) q4; cx q2,q4; u1(pi/4) q4; cx q3,q4; u1(-pi/4) q4; cx q2,q4; u1(pi/4) q4; cx q3,q4; u2(0,pi) q3; rccx q0,q1,q4; }}
+gate mcu1(param0) q0,q1,q2,q3,q4,q5 {{ cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }}
+gate mcx_gray q0,q1,q2,q3,q4,q5 {{ h q5; mcu1(pi) q0,q1,q2,q3,q4,q5; h q5; }}
+gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 {{ mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; }}
+gate mcx_vchain_{0} q0,q1,q2,q3,q4,q5,q6,q7,q8 {{ rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }}
 qreg q[9];
 mcx_gray q[0],q[1],q[2],q[3],q[4],q[5];
 mcx_recursive q[0],q[1],q[2],q[3],q[4],q[5],q[6];
-mcx_vchain q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];\n"""
+mcx_vchain_{0} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];\n""".format(
+            mcx_vchain_id
+        )
 
         self.assertEqual(qc.qasm(), expected_qasm)
 

From 09f08840f1e81bb737b3e8ba6806dcd93fc541f5 Mon Sep 17 00:00:00 2001
From: rafaella-vale <26910380+rafaella-vale@users.noreply.github.com>
Date: Fri, 9 Sep 2022 12:36:28 -0300
Subject: [PATCH 08/17] remove draw

---
 test/python/circuit/test_circuit_qasm.py | 2 --
 1 file changed, 2 deletions(-)

diff --git a/test/python/circuit/test_circuit_qasm.py b/test/python/circuit/test_circuit_qasm.py
index 452a45abd12d..7c277dfe74e6 100644
--- a/test/python/circuit/test_circuit_qasm.py
+++ b/test/python/circuit/test_circuit_qasm.py
@@ -270,8 +270,6 @@ def test_circuit_qasm_with_mcx_gate(self):
         qc = QuantumCircuit(4)
         qc.mcx([0, 1, 2], 3)
 
-        print(qc.decompose().draw())
-
         # qasm output doesn't support parameterized gate yet.
         # param0 for "gate mcuq(param0) is not used inside the definition
         expected_qasm = """OPENQASM 2.0;

From be4feb5ed15c06b6cd09d9c3bef4228b3a222915 Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Mon, 15 Jul 2024 03:01:55 -0500
Subject: [PATCH 09/17] fix qasm file in
 test_circuit_qasm_with_mcx_gate_variants

---
 test/python/circuit/test_circuit_qasm.py | 16 +++++++++-------
 1 file changed, 9 insertions(+), 7 deletions(-)

diff --git a/test/python/circuit/test_circuit_qasm.py b/test/python/circuit/test_circuit_qasm.py
index 89f7d07e4b4e..46d96011c227 100644
--- a/test/python/circuit/test_circuit_qasm.py
+++ b/test/python/circuit/test_circuit_qasm.py
@@ -418,16 +418,18 @@ def test_circuit_qasm_with_mcx_gate_variants(self):
         # param0 for "gate mcuq(param0) is not used inside the definition
         expected_qasm = """OPENQASM 2.0;
 include "qelib1.inc";
-gate mcu1(param0) q0,q1,q2,q3,q4,q5 { cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }
-gate mcx_gray q0,q1,q2,q3,q4,q5 { h q5; mcu1(pi) q0,q1,q2,q3,q4,q5; h q5; }
-gate mcx q0,q1,q2,q3 { h q3; p(pi/8) q0; p(pi/8) q1; p(pi/8) q2; p(pi/8) q3; cx q0,q1; p(-pi/8) q1; cx q0,q1; cx q1,q2; p(-pi/8) q2; cx q0,q2; p(pi/8) q2; cx q1,q2; p(-pi/8) q2; cx q0,q2; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; h q3; }
-gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 { mcx q0,q1,q2,q6; mcx q3,q4,q6,q5; mcx q0,q1,q2,q6; mcx q3,q4,q6,q5; }
-gate mcx_vchain q0,q1,q2,q3,q4,q5,q6,q7,q8 { rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }
+gate mcu1(param0) q0,q1,q2,q3,q4,q5 {{ cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }}
+gate mcx_gray q0,q1,q2,q3,q4,q5 {{ h q5; mcu1(pi) q0,q1,q2,q3,q4,q5; h q5; }}
+gate mcx q0,q1,q2,q3 {{ h q3; p(pi/8) q0; p(pi/8) q1; p(pi/8) q2; p(pi/8) q3; cx q0,q1; p(-pi/8) q1; cx q0,q1; cx q1,q2; p(-pi/8) q2; cx q0,q2; p(pi/8) q2; cx q1,q2; p(-pi/8) q2; cx q0,q2; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; h q3; }}
+gate mcx_vchain q0,q1,q2,q3,q4 {{ mcx q0,q1,q2,q3; }}
+gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 {{ mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; }}
+gate mcx_vchain_{0} q0,q1,q2,q3,q4,q5,q6,q7,q8 {{ rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }}
 qreg q[9];
 mcx_gray q[0],q[1],q[2],q[3],q[4],q[5];
 mcx_recursive q[0],q[1],q[2],q[3],q[4],q[5],q[6];
-mcx_vchain q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];"""
-
+mcx_vchain_{0} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];\n""".format(
+            mcx_vchain_id
+        )
         self.assertEqual(dumps(qc), expected_qasm)
 
     def test_circuit_qasm_with_registerless_bits(self):

From 1507e5cb87d0b1f648d94f60a493a3f8495d3b6a Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Mon, 15 Jul 2024 06:24:55 -0500
Subject: [PATCH 10/17] update the MCXRecursive class docstring

---
 qiskit/circuit/library/standard_gates/x.py | 11 ++++++++---
 1 file changed, 8 insertions(+), 3 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index a6d7f911ef83..b21fd393ff47 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/qiskit/circuit/library/standard_gates/x.py
@@ -1320,9 +1320,14 @@ def _define(self):
 class MCXRecursive(MCXGate):
     """Implement the multi-controlled X gate using recursion.
 
-    Using a single ancilla qubit, the multi-controlled X gate is recursively split onto
-    four sub-registers. This is done until we reach the 3- or 4-controlled X gate since
-    for these we have a concrete implementation that do not require ancillas.
+    Using a single clean ancilla qubit, the multi-controlled X gate is split into
+    four sub-registers, each one of them uses the V-chain method.
+
+    The method is based on Lemma 9 of [2], first shown in Lemma 7.3 of [1].
+
+    References:
+        [1] Barenco et al., 1995. https://arxiv.org/pdf/quant-ph/9503016.pdf
+        [2] Iten et al., 2015. https://arxiv.org/abs/1501.06911
     """
 
     def __init__(

From 76199c942d5f73fea8c24d146c51ee2e9babc536 Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Mon, 15 Jul 2024 07:07:07 -0500
Subject: [PATCH 11/17] add a test of the upper bound limit of the number of CX
 gates

---
 test/python/circuit/test_controlled_gate.py | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

diff --git a/test/python/circuit/test_controlled_gate.py b/test/python/circuit/test_controlled_gate.py
index e36ee04e2162..25fa55b62203 100644
--- a/test/python/circuit/test_controlled_gate.py
+++ b/test/python/circuit/test_controlled_gate.py
@@ -820,6 +820,22 @@ def test_mcxvchain_dirty_ancilla_cx_count(self, num_ctrl_qubits):
 
         self.assertLessEqual(cx_count, 8 * num_ctrl_qubits - 6)
 
+    @data(7, 10, 15)
+    def test_mcxrecursive_clean_ancilla_cx_count(self, num_ctrl_qubits):
+        """Test if cx count of the mcx with one clean ancilla
+        is less than upper bound."""
+        from qiskit import transpile
+
+        mcx_recursive = MCXRecursive(num_ctrl_qubits)
+        qc = QuantumCircuit(mcx_recursive.num_qubits)
+
+        qc.append(mcx_recursive, list(range(mcx_recursive.num_qubits)))
+
+        tr_mcx_rec = transpile(qc, basis_gates=["u", "cx"])
+        cx_count = tr_mcx_rec.count_ops()["cx"]
+
+        self.assertLessEqual(cx_count, 16 * num_ctrl_qubits - 8)
+
     def test_mcxvchain_dirty_ancilla_action_only(self):
         """Test the v-chain mcx with dirty auxiliary qubits
         with gate cancelling with mirrored circuit."""

From 187994f76daa1fd6a6ff5ed27846b53cbe86e689 Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Mon, 15 Jul 2024 08:00:24 -0500
Subject: [PATCH 12/17] fix failing qasm test

---
 test/python/circuit/test_circuit_qasm.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/python/circuit/test_circuit_qasm.py b/test/python/circuit/test_circuit_qasm.py
index 46d96011c227..12bb9cd47016 100644
--- a/test/python/circuit/test_circuit_qasm.py
+++ b/test/python/circuit/test_circuit_qasm.py
@@ -427,7 +427,7 @@ def test_circuit_qasm_with_mcx_gate_variants(self):
 qreg q[9];
 mcx_gray q[0],q[1],q[2],q[3],q[4],q[5];
 mcx_recursive q[0],q[1],q[2],q[3],q[4],q[5],q[6];
-mcx_vchain_{0} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];\n""".format(
+mcx_vchain_{0} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];""".format(
             mcx_vchain_id
         )
         self.assertEqual(dumps(qc), expected_qasm)

From 898b29d6332e665f25161e0f5b5042c720fe998c Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Tue, 16 Jul 2024 02:20:18 -0500
Subject: [PATCH 13/17] fix qasm file in test_export in qasm2 tests

---
 test/python/qasm2/test_export.py | 16 ++++++++++------
 1 file changed, 10 insertions(+), 6 deletions(-)

diff --git a/test/python/qasm2/test_export.py b/test/python/qasm2/test_export.py
index 85172ec3ce8f..1e54f3b3cb11 100644
--- a/test/python/qasm2/test_export.py
+++ b/test/python/qasm2/test_export.py
@@ -402,21 +402,25 @@ def test_mcx_gate_variants(self):
         qc.append(lib.MCXGrayCode(n), range(n + 1))
         qc.append(lib.MCXRecursive(n), range(n + 2))
         qc.append(lib.MCXVChain(n), range(2 * n - 1))
+        mcx_vchain_id = id(qc.data[-1].operation)
 
         # qasm output doesn't support parameterized gate yet.
         # param0 for "gate mcuq(param0) is not used inside the definition
         expected_qasm = """\
 OPENQASM 2.0;
 include "qelib1.inc";
-gate mcu1(param0) q0,q1,q2,q3,q4,q5 { cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }
-gate mcx_gray q0,q1,q2,q3,q4,q5 { h q5; mcu1(pi) q0,q1,q2,q3,q4,q5; h q5; }
-gate mcx q0,q1,q2,q3 { h q3; p(pi/8) q0; p(pi/8) q1; p(pi/8) q2; p(pi/8) q3; cx q0,q1; p(-pi/8) q1; cx q0,q1; cx q1,q2; p(-pi/8) q2; cx q0,q2; p(pi/8) q2; cx q1,q2; p(-pi/8) q2; cx q0,q2; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; h q3; }
-gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 { mcx q0,q1,q2,q6; mcx q3,q4,q6,q5; mcx q0,q1,q2,q6; mcx q3,q4,q6,q5; }
-gate mcx_vchain q0,q1,q2,q3,q4,q5,q6,q7,q8 { rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }
+gate mcu1(param0) q0,q1,q2,q3,q4,q5 {{ cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }}
+gate mcx_gray q0,q1,q2,q3,q4,q5 {{ h q5; mcu1(pi) q0,q1,q2,q3,q4,q5; h q5; }}
+gate mcx q0,q1,q2,q3 {{ h q3; p(pi/8) q0; p(pi/8) q1; p(pi/8) q2; p(pi/8) q3; cx q0,q1; p(-pi/8) q1; cx q0,q1; cx q1,q2; p(-pi/8) q2; cx q0,q2; p(pi/8) q2; cx q1,q2; p(-pi/8) q2; cx q0,q2; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; h q3; }}
+gate mcx_vchain q0,q1,q2,q3,q4 {{ mcx q0,q1,q2,q3; }}
+gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 {{ mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; }}
+gate mcx_vchain_{0} q0,q1,q2,q3,q4,q5,q6,q7,q8 {{ rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }}
 qreg q[9];
 mcx_gray q[0],q[1],q[2],q[3],q[4],q[5];
 mcx_recursive q[0],q[1],q[2],q[3],q[4],q[5],q[6];
-mcx_vchain q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];"""
+mcx_vchain_{0} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];""".format(
+            mcx_vchain_id
+        )
 
         self.assertEqual(qasm2.dumps(qc), expected_qasm)
 

From 11dc6c956aa42b45925dd8abc427bc6ef6cc1623 Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Tue, 16 Jul 2024 02:51:15 -0500
Subject: [PATCH 14/17] fix format lint errors in qasm strings

---
 test/python/circuit/test_circuit_qasm.py | 9 ++++-----
 test/python/qasm2/test_export.py         | 8 +++-----
 2 files changed, 7 insertions(+), 10 deletions(-)

diff --git a/test/python/circuit/test_circuit_qasm.py b/test/python/circuit/test_circuit_qasm.py
index 12bb9cd47016..5f01e5c647a6 100644
--- a/test/python/circuit/test_circuit_qasm.py
+++ b/test/python/circuit/test_circuit_qasm.py
@@ -416,20 +416,19 @@ def test_circuit_qasm_with_mcx_gate_variants(self):
 
         # qasm output doesn't support parameterized gate yet.
         # param0 for "gate mcuq(param0) is not used inside the definition
-        expected_qasm = """OPENQASM 2.0;
+        expected_qasm = f"""OPENQASM 2.0;
 include "qelib1.inc";
 gate mcu1(param0) q0,q1,q2,q3,q4,q5 {{ cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }}
 gate mcx_gray q0,q1,q2,q3,q4,q5 {{ h q5; mcu1(pi) q0,q1,q2,q3,q4,q5; h q5; }}
 gate mcx q0,q1,q2,q3 {{ h q3; p(pi/8) q0; p(pi/8) q1; p(pi/8) q2; p(pi/8) q3; cx q0,q1; p(-pi/8) q1; cx q0,q1; cx q1,q2; p(-pi/8) q2; cx q0,q2; p(pi/8) q2; cx q1,q2; p(-pi/8) q2; cx q0,q2; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; h q3; }}
 gate mcx_vchain q0,q1,q2,q3,q4 {{ mcx q0,q1,q2,q3; }}
 gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 {{ mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; }}
-gate mcx_vchain_{0} q0,q1,q2,q3,q4,q5,q6,q7,q8 {{ rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }}
+gate mcx_vchain_{mcx_vchain_id} q0,q1,q2,q3,q4,q5,q6,q7,q8 {{ rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }}
 qreg q[9];
 mcx_gray q[0],q[1],q[2],q[3],q[4],q[5];
 mcx_recursive q[0],q[1],q[2],q[3],q[4],q[5],q[6];
-mcx_vchain_{0} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];""".format(
-            mcx_vchain_id
-        )
+mcx_vchain_{mcx_vchain_id} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];"""
+
         self.assertEqual(dumps(qc), expected_qasm)
 
     def test_circuit_qasm_with_registerless_bits(self):
diff --git a/test/python/qasm2/test_export.py b/test/python/qasm2/test_export.py
index 1e54f3b3cb11..0bba2f5c47e0 100644
--- a/test/python/qasm2/test_export.py
+++ b/test/python/qasm2/test_export.py
@@ -406,7 +406,7 @@ def test_mcx_gate_variants(self):
 
         # qasm output doesn't support parameterized gate yet.
         # param0 for "gate mcuq(param0) is not used inside the definition
-        expected_qasm = """\
+        expected_qasm = f"""\
 OPENQASM 2.0;
 include "qelib1.inc";
 gate mcu1(param0) q0,q1,q2,q3,q4,q5 {{ cu1(pi/16) q4,q5; cx q4,q3; cu1(-pi/16) q3,q5; cx q4,q3; cu1(pi/16) q3,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q3,q2; cu1(-pi/16) q2,q5; cx q4,q2; cu1(pi/16) q2,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q2,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q3,q1; cu1(-pi/16) q1,q5; cx q4,q1; cu1(pi/16) q1,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q1,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q2,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; cx q3,q0; cu1(-pi/16) q0,q5; cx q4,q0; cu1(pi/16) q0,q5; }}
@@ -414,13 +414,11 @@ def test_mcx_gate_variants(self):
 gate mcx q0,q1,q2,q3 {{ h q3; p(pi/8) q0; p(pi/8) q1; p(pi/8) q2; p(pi/8) q3; cx q0,q1; p(-pi/8) q1; cx q0,q1; cx q1,q2; p(-pi/8) q2; cx q0,q2; p(pi/8) q2; cx q1,q2; p(-pi/8) q2; cx q0,q2; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q1,q3; p(pi/8) q3; cx q2,q3; p(-pi/8) q3; cx q0,q3; h q3; }}
 gate mcx_vchain q0,q1,q2,q3,q4 {{ mcx q0,q1,q2,q3; }}
 gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 {{ mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; mcx_vchain q0,q1,q2,q6,q3; mcx_vchain q3,q4,q6,q5,q0; }}
-gate mcx_vchain_{0} q0,q1,q2,q3,q4,q5,q6,q7,q8 {{ rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }}
+gate mcx_vchain_{mcx_vchain_id} q0,q1,q2,q3,q4,q5,q6,q7,q8 {{ rccx q0,q1,q6; rccx q2,q6,q7; rccx q3,q7,q8; ccx q4,q8,q5; rccx q3,q7,q8; rccx q2,q6,q7; rccx q0,q1,q6; }}
 qreg q[9];
 mcx_gray q[0],q[1],q[2],q[3],q[4],q[5];
 mcx_recursive q[0],q[1],q[2],q[3],q[4],q[5],q[6];
-mcx_vchain_{0} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];""".format(
-            mcx_vchain_id
-        )
+mcx_vchain_{mcx_vchain_id} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];"""
 
         self.assertEqual(qasm2.dumps(qc), expected_qasm)
 

From de55690b03d23e7a45e77f0167fa6685c8ab1c83 Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Tue, 16 Jul 2024 02:51:34 -0500
Subject: [PATCH 15/17] add release notes

---
 .../notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml  | 5 +++++
 1 file changed, 5 insertions(+)
 create mode 100644 releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml

diff --git a/releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml b/releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml
new file mode 100644
index 000000000000..26c1772cef22
--- /dev/null
+++ b/releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml
@@ -0,0 +1,5 @@
+---
+features:
+  - |
+    :class:`.MCXRecursive` with k controls and a single clean auxiliary qubits now requires at most
+    16*k-8 cx gates.

From ee273193cd66f470e3b0d2fd9e3987023175d22a Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Tue, 16 Jul 2024 10:44:39 -0500
Subject: [PATCH 16/17] update references format

---
 qiskit/circuit/library/standard_gates/x.py | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/qiskit/circuit/library/standard_gates/x.py b/qiskit/circuit/library/standard_gates/x.py
index b21fd393ff47..3688d376538a 100644
--- a/qiskit/circuit/library/standard_gates/x.py
+++ b/qiskit/circuit/library/standard_gates/x.py
@@ -952,8 +952,8 @@ class C4XGate(SingletonControlledGate):
     of the relative phase version of c3x, the rc3x [2].
 
     References:
-        [1] Barenco et al., 1995. https://arxiv.org/pdf/quant-ph/9503016.pdf
-        [2] Maslov, 2015. https://arxiv.org/abs/1508.03273
+        1. Barenco et al., 1995. https://arxiv.org/pdf/quant-ph/9503016.pdf
+        2. Maslov, 2015. https://arxiv.org/abs/1508.03273
     """
 
     def __init__(
@@ -1326,8 +1326,8 @@ class MCXRecursive(MCXGate):
     The method is based on Lemma 9 of [2], first shown in Lemma 7.3 of [1].
 
     References:
-        [1] Barenco et al., 1995. https://arxiv.org/pdf/quant-ph/9503016.pdf
-        [2] Iten et al., 2015. https://arxiv.org/abs/1501.06911
+        1. Barenco et al., 1995. https://arxiv.org/pdf/quant-ph/9503016.pdf
+        2. Iten et al., 2015. https://arxiv.org/abs/1501.06911
     """
 
     def __init__(

From e78a4c67d60ec1a625d7b0ab198109e83a9d9c15 Mon Sep 17 00:00:00 2001
From: ShellyGarion <shelly@il.ibm.com>
Date: Wed, 17 Jul 2024 06:55:08 -0500
Subject: [PATCH 17/17] update release notes after review

---
 .../notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml b/releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml
index 26c1772cef22..12ccd09f6e12 100644
--- a/releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml
+++ b/releasenotes/notes/mcx_recursive_clean_ancilla-2b0f6956e0f4cbbd.yaml
@@ -1,5 +1,5 @@
 ---
-features:
+features_synthesis:
   - |
-    :class:`.MCXRecursive` with k controls and a single clean auxiliary qubits now requires at most
-    16*k-8 cx gates.
+    :class:`.MCXRecursive` with :math:`k` control qubits and a single clean auxiliary qubit
+    now requires at most :math:`16k-8` CX gates.