Minor fixes/improvements to some MCX synthesis methods (#14093)

* updates to synth_mcx_n_dirty_i15 and to synth_mcx_1_clean_b95

* add explicit tests for MCX synthesis algorithms

* improving comment

* rename variable

* slightly generalize and simplify _msu_real_diagonal

* updating qasm tests

* improving treatment of special cases in synth_mcx_n_dirty_i15

* remove duplicated tests

* add tests for counting CX gates

* add some more cx_count tests

* remove tests

* improving docstring comments (minor)

* pass over tests

* docstring

* increasing the number of control qubits in some tests

* changing circuit's name to mcx_vchain for compatibility

* cleanup

* fixing MCX qasms

* updating docstrings

* reno

* tests and docs

* fix qasm tests

* addressing review comments

* futher expanding review comment

---------

Co-authored-by: ShellyGarion <shelly@il.ibm.com>

Author: alexanderivrii

qiskit/synthesis/multi_controlled/mcx_synthesis.py

@@ -32,9 +32,12 @@ def synth_mcx_n_dirty_i15(
     action_only: bool = False,
 ) -> QuantumCircuit:
     r"""
-    Synthesize a multi-controlled X gate with :math:`k` controls using :math:`k - 2`
-    dirty ancillary qubits producing a circuit with :math:`2 * k - 1` qubits and at most
-    :math:`8 * k - 6` CX gates, by Iten et. al. [1].
+    Synthesize a multi-controlled X gate with :math:`k` controls based on the paper
+    by Iten et al. [1].
+
+    For :math:`k\ge 4` the method uses :math:`k - 2` dirty ancillary qubits, producing a circuit
+    with :math:`2 * k - 1` qubits and at most :math:`8 * k - 6` CX gates. For :math:`k\le 3`
+    explicit efficient circuits are used instead.
 
     Args:
         num_ctrl_qubits: The number of control qubits.
@@ -53,27 +56,26 @@ def synth_mcx_n_dirty_i15(
            `arXiv:1501.06911 <http://arxiv.org/abs/1501.06911>`_
     """
 
+    # First, handle some special cases
     if num_ctrl_qubits == 1:
-        num_qubits = 2
-    else:
-        num_qubits = 2 * num_ctrl_qubits - 1
-    q = QuantumRegister(num_qubits, name="q")
-    qc = QuantumCircuit(q, name="mcx_vchain")
-    q_controls = q[:num_ctrl_qubits]
-    q_target = q[num_ctrl_qubits]
-    q_ancillas = q[num_ctrl_qubits + 1 :]
-
-    if num_ctrl_qubits == 1:
-        qc.cx(q_controls, q_target)
+        qc = QuantumCircuit(2)
+        qc.cx(0, 1)
         return qc
     elif num_ctrl_qubits == 2:
-        qc.ccx(q_controls[0], q_controls[1], q_target)
+        qc = QuantumCircuit(3)
+        qc.ccx(0, 1, 2)
         return qc
-    elif not relative_phase and num_ctrl_qubits == 3:
-        circuit = synth_c3x()
-        qc.compose(circuit, [*q_controls, q_target], inplace=True, copy=False)
+    elif num_ctrl_qubits == 3 and not relative_phase:
+        qc = synth_c3x()
+        qc.name = "mcx_vchain"
         return qc
 
+    num_qubits = 2 * num_ctrl_qubits - 1
+    q = QuantumRegister(num_qubits, name="q")
+    qc = QuantumCircuit(q, name="mcx_vchain")
+    q_controls = q[:num_ctrl_qubits]
+    q_target = q[num_ctrl_qubits]
+    q_ancillas = q[num_ctrl_qubits + 1 :]
     num_ancillas = num_ctrl_qubits - 2
     targets = [q_target] + q_ancillas[:num_ancillas][::-1]
 
@@ -181,7 +183,7 @@ def synth_mcx_1_clean_b95(num_ctrl_qubits: int) -> QuantumCircuit:
     r"""
     Synthesize a multi-controlled X gate with :math:`k` controls using a single
     clean ancillary qubit producing a circuit with :math:`k + 2` qubits and at most
-    :math:`16 * k - 8` CX gates, by Barenco et al. [1].
+    :math:`16 * k - 24` CX gates, by [1], [2].
 
     Args:
         num_ctrl_qubits: The number of control qubits.
@@ -190,8 +192,10 @@ def synth_mcx_1_clean_b95(num_ctrl_qubits: int) -> QuantumCircuit:
         The synthesized quantum circuit.
 
     References:
-        1. Barenco et. al., Phys.Rev. A52 3457 (1995),
+        1. Barenco et. al., *Elementary gates for quantum computation*, Phys.Rev. A52 3457 (1995),
            `arXiv:quant-ph/9503016 <https://arxiv.org/abs/quant-ph/9503016>`_
+        2. Iten et. al., *Quantum Circuits for Isometries*, Phys. Rev. A 93, 032318 (2016),
+           `arXiv:1501.06911 <http://arxiv.org/abs/1501.06911>`_
     """
 
     if num_ctrl_qubits == 3:
@@ -208,32 +212,27 @@ def synth_mcx_1_clean_b95(num_ctrl_qubits: int) -> QuantumCircuit:
     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_first_half = synth_mcx_n_dirty_i15(num_ctrl_qubits=len(first_half))
-    qc_second_half = synth_mcx_n_dirty_i15(num_ctrl_qubits=len(second_half))
-
-    qc.append(
-        qc_first_half,
-        qargs=[*first_half, q_ancilla, *q[middle : middle + len(first_half) - 2]],
-        cargs=[],
-    )
-    qc.append(
-        qc_second_half,
-        qargs=[*second_half, q_target, *q[: len(second_half) - 2]],
-        cargs=[],
-    )
-    qc.append(
-        qc_first_half,
-        qargs=[*first_half, q_ancilla, *q[middle : middle + len(first_half) - 2]],
-        cargs=[],
-    )
-    qc.append(
-        qc_second_half,
-        qargs=[*second_half, q_target, *q[: len(second_half) - 2]],
-        cargs=[],
-    )
+    # The contruction involving 4 MCX gates is described in Lemma 7.3 of [1], and also
+    # appears as Lemma 9 in [2]. The optimization that the first and third MCX gates
+    # can be synthesized up to relative phase follows from Lemma 7 in [2], as a diagonal
+    # gate following the first MCX gate commutes with the second MCX gate, and
+    # thus cancels with the inverse diagonal gate preceding the third MCX gate. The
+    # same optimization cannot be applied to the second MCX gate, since a diagonal
+    # gate following the second MCX gate would not satisfy the preconditions of Lemma 7,
+    # and would not necessarily commute with the third MCX gate.
+    controls1 = [*q[:middle]]
+    mcx1 = synth_mcx_n_dirty_i15(num_ctrl_qubits=len(controls1), relative_phase=True)
+    qubits1 = [*controls1, q_ancilla, *q[middle : middle + mcx1.num_qubits - len(controls1) - 1]]
+
+    controls2 = [*q[middle : num_ctrl_qubits - 1], q_ancilla]
+    mcx2 = synth_mcx_n_dirty_i15(num_ctrl_qubits=len(controls2))
+    qc2_qubits = [*controls2, q_target, *q[0 : mcx2.num_qubits - len(controls2) - 1]]
+
+    qc.compose(mcx1, qubits1, inplace=True)
+    qc.compose(mcx2, qc2_qubits, inplace=True)
+    qc.compose(mcx1.inverse(), qubits1, inplace=True)
+    qc.compose(mcx2, qc2_qubits, inplace=True)
 
     return qc
 

qiskit/synthesis/multi_controlled/multi_control_rotation_gates.py

@@ -171,30 +171,29 @@ def _mcsu2_real_diagonal(
     k_1 = math.ceil(num_controls / 2.0)
     k_2 = math.floor(num_controls / 2.0)
 
-    circuit = QuantumCircuit(num_controls + 1, name="MCSU2")
     controls = list(range(num_controls))  # control indices, defined for code legibility
     target = num_controls  # target index, defined for code legibility
 
+    mcx1 = synth_mcx_n_dirty_i15(num_ctrl_qubits=k_1)
+    mcx1_num_ancillas = mcx1.num_qubits - k_1 - 1
+    mcx1_qubits = controls[:k_1] + [target] + controls[k_1 : k_1 + mcx1_num_ancillas]
+
+    mcx2 = synth_mcx_n_dirty_i15(num_ctrl_qubits=k_2)
+    mcx2_num_ancillas = mcx2.num_qubits - k_2 - 1
+    mcx2_qubits = controls[k_1:] + [target] + controls[k_1 - mcx2_num_ancillas : k_1]
+
+    circuit = QuantumCircuit(num_controls + 1, name="MCSU2")
+
     if not is_secondary_diag_real:
         circuit.h(target)
 
-    mcx_1 = synth_mcx_n_dirty_i15(num_ctrl_qubits=k_1)
-    circuit.compose(mcx_1, controls[:k_1] + [target] + controls[k_1 : 2 * k_1 - 2], inplace=True)
+    circuit.compose(mcx1, mcx1_qubits, inplace=True)
     circuit.append(s_gate, [target])
-
-    # TODO: improve CX count by using action_only=True (based on #9687)
-    mcx_2 = synth_mcx_n_dirty_i15(num_ctrl_qubits=k_2).to_gate()
-    circuit.compose(
-        mcx_2.inverse(), controls[k_1:] + [target] + controls[k_1 - k_2 + 2 : k_1], inplace=True
-    )
+    circuit.compose(mcx2, mcx2_qubits, inplace=True)
     circuit.append(s_gate.inverse(), [target])
-
-    mcx_3 = synth_mcx_n_dirty_i15(num_ctrl_qubits=k_1).to_gate()
-    circuit.compose(mcx_3, controls[:k_1] + [target] + controls[k_1 : 2 * k_1 - 2], inplace=True)
+    circuit.compose(mcx1, mcx1_qubits, inplace=True)
     circuit.append(s_gate, [target])
-
-    mcx_4 = synth_mcx_n_dirty_i15(num_ctrl_qubits=k_2).to_gate()
-    circuit.compose(mcx_4, controls[k_1:] + [target] + controls[k_1 - k_2 + 2 : k_1], inplace=True)
+    circuit.compose(mcx2, mcx2_qubits, inplace=True)
     circuit.append(s_gate.inverse(), [target])
 
     if not is_secondary_diag_real:

qiskit/transpiler/passes/synthesis/hls_plugins.py

@@ -1008,9 +1008,10 @@ class MCXSynthesisNDirtyI15(HighLevelSynthesisPlugin):
     This plugin name is :``mcx.n_dirty_i15`` which can be used as the key on
     an :class:`~.HLSConfig` object to use this method with :class:`~.HighLevelSynthesis`.
 
-    For a multi-controlled X gate with :math:`k\ge 3` control qubits this synthesis
+    For a multi-controlled X gate with :math:`k\ge 4` control qubits this synthesis
     method requires :math:`k - 2` additional dirty auxiliary qubits. The synthesized
     circuit consists of :math:`2 * k - 1` qubits and at most :math:`8 * k - 6` CX gates.
+    For :math:`k\le 3` explicit efficient circuits are used instead.
 
     The plugin supports the following plugin-specific options:
 
@@ -1105,14 +1106,14 @@ class MCXSynthesis1CleanB95(HighLevelSynthesisPlugin):
 
     For a multi-controlled X gate with :math:`k\ge 5` control qubits this synthesis
     method requires a single additional clean auxiliary qubit. The synthesized
-    circuit consists of :math:`k + 2` qubits and at most :math:`16 * k - 8` CX gates.
+    circuit consists of :math:`k + 2` qubits and at most :math:`16 * k - 24` CX gates.
 
     The plugin supports the following plugin-specific options:
 
     * num_clean_ancillas: The number of clean auxiliary qubits available.
 
     References:
-        1. Barenco et. al., Phys.Rev. A52 3457 (1995),
+        1. Barenco et. al., *Elementary gates for quantum computation*, Phys.Rev. A52 3457 (1995),
            `arXiv:quant-ph/9503016 <https://arxiv.org/abs/quant-ph/9503016>`_
     """
 

releasenotes/notes/minor-mcx-improvements-a1ed81f83275ce35.yaml

@@ -0,0 +1,18 @@
+---
+features_synthesis:
+  - |
+    The synthesis function :func:`.synth_mcx_1_clean_b95` now produces a circuit
+    with fewer CX-gates.
+upgrade_synthesis:
+  - |
+    The synthesis function :func:`.synth_mcx_1_clean_b95` now produces a circuit
+    with fewer layers of wrappings.
+fixes:
+  - |
+    When synthesizing an :class:`.MCXGate` gate with 3 controls, the synthesis
+    functon :func:`.synth_mcx_n_dirty_i15` used to require one auxiliary qubit,
+    producing a circuit with 5 qubits (3 control, 1 target, and 1 auxiliary). 
+    However, the actual synthesis algorithm does not make use of this auxiliary
+    qubit. This behavior is now fixed: the synthesized circuit is over 4 qubits
+    (3 control and 1 target), allowing to apply the synthesis function in a
+    slighty larger number of cases.

test/python/circuit/test_circuit_qasm.py

@@ -406,20 +406,16 @@ 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 = f"""OPENQASM 2.0;
+        expected_qasm = """OPENQASM 2.0;
 include "qelib1.inc";
-gate mcx_gray q0,q1,q2,q3,q4,q5 {{ h 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; h q5; }}
-gate mcx_vchain q0,q1,q2,q3,q4 {{ 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_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_{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; }}
+gate mcx_gray q0,q1,q2,q3,q4,q5 { h 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; h q5; }
+gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 { h q6; t q6; cx q2,q6; tdg q6; cx q3,q6; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; cx q3,q6; t q6; cx q2,q6; tdg q6; h q6; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; h q5; p(pi/8) q3; p(pi/8) q4; p(pi/8) q6; p(pi/8) q5; cx q3,q4; p(-pi/8) q4; cx q3,q4; cx q4,q6; p(-pi/8) q6; cx q3,q6; p(pi/8) q6; cx q4,q6; p(-pi/8) q6; cx q3,q6; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; h q5; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; h q6; t q6; cx q2,q6; tdg q6; cx q3,q6; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; cx q3,q6; t q6; cx q2,q6; tdg q6; h q6; h q5; p(pi/8) q3; p(pi/8) q4; p(pi/8) q6; p(pi/8) q5; cx q3,q4; p(-pi/8) q4; cx q3,q4; cx q4,q6; p(-pi/8) q6; cx q3,q6; p(pi/8) q6; cx q4,q6; p(-pi/8) q6; cx q3,q6; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; h 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; }
 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_{mcx_vchain_id} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];"""
+mcx_vchain q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];"""
 
         self.assertEqual(dumps(qc), expected_qasm)
 

test/python/qasm2/test_export.py

@@ -396,20 +396,16 @@ 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 = f"""OPENQASM 2.0;
+        expected_qasm = """OPENQASM 2.0;
 include "qelib1.inc";
-gate mcx_gray q0,q1,q2,q3,q4,q5 {{ h 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; h q5; }}
-gate mcx_vchain q0,q1,q2,q3,q4 {{ 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_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_{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; }}
+gate mcx_gray q0,q1,q2,q3,q4,q5 { h 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; h q5; }
+gate mcx_recursive q0,q1,q2,q3,q4,q5,q6 { h q6; t q6; cx q2,q6; tdg q6; cx q3,q6; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; cx q3,q6; t q6; cx q2,q6; tdg q6; h q6; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; h q5; p(pi/8) q3; p(pi/8) q4; p(pi/8) q6; p(pi/8) q5; cx q3,q4; p(-pi/8) q4; cx q3,q4; cx q4,q6; p(-pi/8) q6; cx q3,q6; p(pi/8) q6; cx q4,q6; p(-pi/8) q6; cx q3,q6; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; h q5; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; h q6; t q6; cx q2,q6; tdg q6; cx q3,q6; h q3; t q3; cx q0,q3; tdg q3; cx q1,q3; t q3; cx q0,q3; tdg q3; h q3; cx q3,q6; t q6; cx q2,q6; tdg q6; h q6; h q5; p(pi/8) q3; p(pi/8) q4; p(pi/8) q6; p(pi/8) q5; cx q3,q4; p(-pi/8) q4; cx q3,q4; cx q4,q6; p(-pi/8) q6; cx q3,q6; p(pi/8) q6; cx q4,q6; p(-pi/8) q6; cx q3,q6; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q4,q5; p(pi/8) q5; cx q6,q5; p(-pi/8) q5; cx q3,q5; h 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; }
 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_{mcx_vchain_id} q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8];"""
+mcx_vchain 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)
 

test/python/synthesis/test_multi_controlled_synthesis.py

@@ -270,7 +270,7 @@ def test_mcx_1_clean_b95_cx_count(self, num_ctrl_qubits: int):
         transpiled_circuit = self.pm.run(synthesized_circuit)
         cx_count = transpiled_circuit.count_ops()["cx"]
         # The bound from the documentation of synth_mcx_1_clean_b95
-        self.assertLessEqual(cx_count, 16 * num_ctrl_qubits - 8)
+        self.assertLessEqual(cx_count, 16 * num_ctrl_qubits - 24)
 
     @data(3, 5, 10, 15)
     def test_mcx_1_clean_kg24_cx_count(self, num_ctrl_qubits: int):