text: # This code is part of Qiskit.
  #
  # (C) Copyright IBM 2017.
  #
  # This code is licensed under the Apache License, Version 2.0. You may
  # obtain a copy of this license in the LICENSE.txt file in the root directory
  # of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
  #
  # Any modifications or derivative works of this code must retain this
  # copyright notice, and modified files need to carry a notice indicating
  # that they have been altered from the originals.
  
  """X, CX, CCX and multi-controlled X gates."""
  
  from math import ceil
  import numpy
  from qiskit.circuit.controlledgate import ControlledGate
  from qiskit.circuit.gate import Gate
  from qiskit.circuit.quantumregister import QuantumRegister
  from qiskit.circuit._utils import _compute_control_matrix, _ctrl_state_to_int
  from qiskit.qasm import pi
  from .h import HGate
  from .t import TGate, TdgGate
  from .u1 import U1Gate
  from .u2 import U2Gate
  
  
  class XGate(Gate):
      r"""The single-qubit Pauli-X gate (:math:`\sigma_x`).
  
      **Matrix Representation:**
  
      .. math::
  
          X = \begin{pmatrix}
                  0 & 1 \\
                  1 & 0
              \end{pmatrix}
  
      **Circuit symbol:**
  
      .. parsed-literal::
  
               ┌───┐
          q_0: ┤ X ├
               └───┘
  
      Equivalent to a :math:`\pi` radian rotation about the X axis.
  
      .. note::
  
          A global phase difference exists between the definitions of
          :math:`RX(\pi)` and :math:`X`.
  
          .. math::
  
              RX(\pi) = \begin{pmatrix}
                          0 & -i \\
                          -i & 0
                        \end{pmatrix}
                      = -i X
  
      The gate is equivalent to a classical bit flip.
  
      .. math::
  
          |0\rangle \rightarrow |1\rangle \\
          |1\rangle \rightarrow |0\rangle
      """
  
      def __init__(self, label=None):
          """Create new X gate."""
          super().__init__('x', 1, [], label=label)
  
      def _define(self):
          """
          gate x a { u3(pi,0,pi) a; }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .u3 import U3Gate
          q = QuantumRegister(1, 'q')
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (U3Gate(pi, 0, pi), [q[0]], [])
          ]
          for instr, qargs, cargs in rules:
              qc._append(instr, qargs, cargs)
  
          self.definition = qc
  
      def control(self, num_ctrl_qubits=1, label=None, ctrl_state=None):
          """Return a (mutli-)controlled-X gate.
  
          One control returns a CX gate. Two controls returns a CCX gate.
  
          Args:
              num_ctrl_qubits (int): number of control qubits.
              label (str or None): An optional label for the gate [Default: None]
              ctrl_state (int or str or None): control state expressed as integer,
                  string (e.g. '110'), or None. If None, use all 1s.
  
          Returns:
              ControlledGate: controlled version of this gate.
          """
          gate = MCXGate(num_ctrl_qubits=num_ctrl_qubits, label=label, ctrl_state=ctrl_state)
          gate.base_gate.label = self.label
          return gate
  
      def inverse(self):
          r"""Return inverted X gate (itself)."""
          return XGate()  # self-inverse
  
      def __array__(self, dtype=None):
          """Return a numpy.array for the X gate."""
          return numpy.array([[0, 1],
                              [1, 0]], dtype=dtype)
  
  
  class CXGate(ControlledGate):
      r"""Controlled-X gate.
  
      **Circuit symbol:**
  
      .. parsed-literal::
  
          q_0: ──■──
               ┌─┴─┐
          q_1: ┤ X ├
               └───┘
  
      **Matrix representation:**
  
      .. math::
  
          CX\ q_0, q_1 =
              I \otimes |0\rangle\langle0| + X \otimes |1\rangle\langle1| =
              \begin{pmatrix}
                  1 & 0 & 0 & 0 \\
                  0 & 0 & 0 & 1 \\
                  0 & 0 & 1 & 0 \\
                  0 & 1 & 0 & 0
              \end{pmatrix}
  
      .. note::
  
          In Qiskit's convention, higher qubit indices are more significant
          (little endian convention). In many textbooks, controlled gates are
          presented with the assumption of more significant qubits as control,
          which in our case would be q_1. Thus a textbook matrix for this
          gate will be:
  
          .. parsed-literal::
                   ┌───┐
              q_0: ┤ X ├
                   └─┬─┘
              q_1: ──■──
  
          .. math::
  
              CX\ q_1, q_0 =
                  |0 \rangle\langle 0| \otimes I + |1 \rangle\langle 1| \otimes X =
                  \begin{pmatrix}
                      1 & 0 & 0 & 0 \\
                      0 & 1 & 0 & 0 \\
                      0 & 0 & 0 & 1 \\
                      0 & 0 & 1 & 0
                  \end{pmatrix}
  
  
      In the computational basis, this gate flips the target qubit
      if the control qubit is in the :math:`|1\rangle` state.
      In this sense it is similar to a classical XOR gate.
  
      .. math::
          `|a, b\rangle \rightarrow |a, a \oplus b\rangle`
      """
  
      def __init__(self, label=None, ctrl_state=None):
          """Create new CX gate."""
          super().__init__('cx', 2, [], num_ctrl_qubits=1, label=label,
                           ctrl_state=ctrl_state, base_gate=XGate())
  
      def control(self, num_ctrl_qubits=1, label=None, ctrl_state=None):
          """Return a controlled-X gate with more control lines.
  
          Args:
              num_ctrl_qubits (int): number of control qubits.
              label (str or None): An optional label for the gate [Default: None]
              ctrl_state (int or str or None): control state expressed as integer,
                  string (e.g. '110'), or None. If None, use all 1s.
  
          Returns:
              ControlledGate: controlled version of this gate.
          """
          ctrl_state = _ctrl_state_to_int(ctrl_state, num_ctrl_qubits)
          new_ctrl_state = (self.ctrl_state << num_ctrl_qubits) | ctrl_state
          gate = MCXGate(num_ctrl_qubits=num_ctrl_qubits + 1, label=label, ctrl_state=new_ctrl_state)
          gate.base_gate.label = self.label
          return gate
  
      def inverse(self):
          """Return inverted CX gate (itself)."""
          return CXGate(ctrl_state=self.ctrl_state)  # self-inverse
  
      def __array__(self, dtype=None):
          """Return a numpy.array for the CX gate."""
          if self.ctrl_state:
              return numpy.array([[1, 0, 0, 0],
                                  [0, 0, 0, 1],
                                  [0, 0, 1, 0],
                                  [0, 1, 0, 0]], dtype=dtype)
          else:
              return numpy.array([[0, 0, 1, 0],
                                  [0, 1, 0, 0],
                                  [1, 0, 0, 0],
                                  [0, 0, 0, 1]], dtype=dtype)
  
  
  class CCXGate(ControlledGate):
      r"""CCX gate, also known as Toffoli gate.
  
      **Circuit symbol:**
  
      .. parsed-literal::
  
          q_0: ──■──
                 │
          q_1: ──■──
               ┌─┴─┐
          q_2: ┤ X ├
               └───┘
  
      **Matrix representation:**
  
      .. math::
  
          CCX q_0, q_1, q_2 =
              I \otimes I \otimes |0 \rangle \langle 0| + CX \otimes |1 \rangle \langle 1| =
             \begin{pmatrix}
                  1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
                  0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
                  0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
                  0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
                  0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
                  0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
                  0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
                  0 & 0 & 0 & 1 & 0 & 0 & 0 & 0
              \end{pmatrix}
  
      .. note::
  
          In Qiskit's convention, higher qubit indices are more significant
          (little endian convention). In many textbooks, controlled gates are
          presented with the assumption of more significant qubits as control,
          which in our case would be q_2 and q_1. Thus a textbook matrix for this
          gate will be:
  
          .. parsed-literal::
                   ┌───┐
              q_0: ┤ X ├
                   └─┬─┘
              q_1: ──■──
                     │
              q_2: ──■──
  
          .. math::
  
              CCX\ q_2, q_1, q_0 =
                  |0 \rangle \langle 0| \otimes I \otimes I + |1 \rangle \langle 1| \otimes CX =
                  \begin{pmatrix}
                      1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
                      0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
                      0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
                      0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
                      0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
                      0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
                      0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
                      0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
                  \end{pmatrix}
  
      """
  
      def __init__(self, label=None, ctrl_state=None):
          """Create new CCX gate."""
          super().__init__('ccx', 3, [], num_ctrl_qubits=2, label=label,
                           ctrl_state=ctrl_state, base_gate=XGate())
  
      def _define(self):
          """
          gate ccx a,b,c
          {
          h c; cx b,c; tdg c; cx a,c;
          t c; cx b,c; tdg c; cx a,c;
          t b; t c; h c; cx a,b;
          t a; tdg b; cx a,b;}
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          q = QuantumRegister(3, 'q')
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (HGate(), [q[2]], []),
              (CXGate(), [q[1], q[2]], []),
              (TdgGate(), [q[2]], []),
              (CXGate(), [q[0], q[2]], []),
              (TGate(), [q[2]], []),
              (CXGate(), [q[1], q[2]], []),
              (TdgGate(), [q[2]], []),
              (CXGate(), [q[0], q[2]], []),
              (TGate(), [q[1]], []),
              (TGate(), [q[2]], []),
              (HGate(), [q[2]], []),
              (CXGate(), [q[0], q[1]], []),
              (TGate(), [q[0]], []),
              (TdgGate(), [q[1]], []),
              (CXGate(), [q[0], q[1]], [])
          ]
          for instr, qargs, cargs in rules:
              qc._append(instr, qargs, cargs)
  
          self.definition = qc
  
      def control(self, num_ctrl_qubits=1, label=None, ctrl_state=None):
          """Controlled version of this gate.
  
          Args:
              num_ctrl_qubits (int): number of control qubits.
              label (str or None): An optional label for the gate [Default: None]
              ctrl_state (int or str or None): control state expressed as integer,
                  string (e.g. '110'), or None. If None, use all 1s.
  
          Returns:
              ControlledGate: controlled version of this gate.
          """
          ctrl_state = _ctrl_state_to_int(ctrl_state, num_ctrl_qubits)
          new_ctrl_state = (self.ctrl_state << num_ctrl_qubits) | ctrl_state
          gate = MCXGate(num_ctrl_qubits=num_ctrl_qubits + 2, label=label, ctrl_state=new_ctrl_state)
          gate.base_gate.label = self.label
          return gate
  
      def inverse(self):
          """Return an inverted CCX gate (also a CCX)."""
          return CCXGate(ctrl_state=self.ctrl_state)  # self-inverse
  
      def __array__(self, dtype=None):
          """Return a numpy.array for the CCX gate."""
          mat = _compute_control_matrix(self.base_gate.to_matrix(),
                                        self.num_ctrl_qubits,
                                        ctrl_state=self.ctrl_state)
          if dtype:
              return numpy.asarray(mat, dtype=dtype)
          return mat
  
  
  class RCCXGate(Gate):
      """The simplified Toffoli gate, also referred to as Margolus gate.
  
      The simplified Toffoli gate implements the Toffoli gate up to relative phases.
      This implementation requires three CX gates which is the minimal amount possible,
      as shown in https://arxiv.org/abs/quant-ph/0312225.
      Note, that the simplified Toffoli is not equivalent to the Toffoli. But can be used in places
      where the Toffoli gate is uncomputed again.
  
      This concrete implementation is from https://arxiv.org/abs/1508.03273, the dashed box
      of Fig. 3.
      """
  
      def __init__(self, label=None):
          """Create a new simplified CCX gate."""
          super().__init__('rccx', 3, [], label=label)
  
      def _define(self):
          """
          gate rccx a,b,c
          { u2(0,pi) c;
            u1(pi/4) c;
            cx b, c;
            u1(-pi/4) c;
            cx a, c;
            u1(pi/4) c;
            cx b, c;
            u1(-pi/4) c;
            u2(0,pi) c;
          }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          q = QuantumRegister(3, 'q')
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (U2Gate(0, pi), [q[2]], []),  # H gate
              (U1Gate(pi / 4), [q[2]], []),  # T gate
              (CXGate(), [q[1], q[2]], []),
              (U1Gate(-pi / 4), [q[2]], []),  # inverse T gate
              (CXGate(), [q[0], q[2]], []),
              (U1Gate(pi / 4), [q[2]], []),
              (CXGate(), [q[1], q[2]], []),
              (U1Gate(-pi / 4), [q[2]], []),  # inverse T gate
              (U2Gate(0, pi), [q[2]], []),  # H gate
          ]
          for instr, qargs, cargs in rules:
              qc._append(instr, qargs, cargs)
  
          self.definition = qc
  
      def __array__(self, dtype=None):
          """Return a numpy.array for the simplified CCX gate."""
          return numpy.array([[1, 0, 0, 0, 0, 0, 0, 0],
                              [0, 1, 0, 0, 0, 0, 0, 0],
                              [0, 0, 1, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, -1j],
                              [0, 0, 0, 0, 1, 0, 0, 0],
                              [0, 0, 0, 0, 0, -1, 0, 0],
                              [0, 0, 0, 0, 0, 0, 1, 0],
                              [0, 0, 0, 1j, 0, 0, 0, 0]], dtype=dtype)
  
  
  class C3XGate(ControlledGate):
      """The 3-qubit controlled X gate.
  
      This implementation is based on Page 17 of [1].
  
      References:
          [1] Barenco et al., 1995. https://arxiv.org/pdf/quant-ph/9503016.pdf
      """
  
      def __init__(self, angle=numpy.pi/4, label=None, ctrl_state=None):
          """Create a new 3-qubit controlled X gate.
  
          Args:
              angle (float): The angle used in the controlled-U1 gates. An angle of π/4 yields the
                  3-qubit controlled X gate, an angle of π/8 the 3-qubit controlled sqrt(X) gate.
              label (str or None): An optional label for the gate [Default: None]
              ctrl_state (int or str or None): control state expressed as integer,
                  string (e.g. '110'), or None. If None, use all 1s.
          """
          super().__init__('mcx', 4, [], num_ctrl_qubits=3, label=label,
                           ctrl_state=ctrl_state, base_gate=XGate())
  
          self._angle = angle
  
      def _define(self):
          """
          gate c3x a,b,c,d
          {
              h d; cu1(-pi/4) a,d; h d;
              cx a,b;
              h d; cu1(pi/4) b,d; h d;
              cx a,b;
              h d; cu1(-pi/4) b,d; h d;
              cx b,c;
              h d; cu1(pi/4) c,d; h d;
              cx a,c;
              h d; cu1(-pi/4) c,d; h d;
              cx b,c;
              h d; cu1(pi/4) c,d; h d;
              cx a,c;
              h d; cu1(-pi/4) c,d; h d;
          }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .u1 import CU1Gate
          q = QuantumRegister(4, name='q')
          rules = [
              (HGate(), [q[3]], []),
              (CU1Gate(-self._angle), [q[0], q[3]], []),
              (HGate(), [q[3]], []),
              (CXGate(), [q[0], q[1]], []),
              (HGate(), [q[3]], []),
              (CU1Gate(self._angle), [q[1], q[3]], []),
              (HGate(), [q[3]], []),
              (CXGate(), [q[0], q[1]], []),
              (HGate(), [q[3]], []),
              (CU1Gate(-self._angle), [q[1], q[3]], []),
              (HGate(), [q[3]], []),
              (CXGate(), [q[1], q[2]], []),
              (HGate(), [q[3]], []),
              (CU1Gate(self._angle), [q[2], q[3]], []),
              (HGate(), [q[3]], []),
              (CXGate(), [q[0], q[2]], []),
              (HGate(), [q[3]], []),
              (CU1Gate(-self._angle), [q[2], q[3]], []),
              (HGate(), [q[3]], []),
              (CXGate(), [q[1], q[2]], []),
              (HGate(), [q[3]], []),
              (CU1Gate(self._angle), [q[2], q[3]], []),
              (HGate(), [q[3]], []),
              (CXGate(), [q[0], q[2]], []),
              (HGate(), [q[3]], []),
              (CU1Gate(-self._angle), [q[2], q[3]], []),
              (HGate(), [q[3]], [])
          ]
          qc = QuantumCircuit(q)
          for instr, qargs, cargs in rules:
              qc._append(instr, qargs, cargs)
  
          self.definition = qc
  
      def control(self, num_ctrl_qubits=1, label=None, ctrl_state=None):
          """Controlled version of this gate.
  
          Args:
              num_ctrl_qubits (int): number of control qubits.
              label (str or None): An optional label for the gate [Default: None]
              ctrl_state (int or str or None): control state expressed as integer,
                  string (e.g. '110'), or None. If None, use all 1s.
  
          Returns:
              ControlledGate: controlled version of this gate.
          """
          ctrl_state = _ctrl_state_to_int(ctrl_state, num_ctrl_qubits)
          new_ctrl_state = (self.ctrl_state << num_ctrl_qubits) | ctrl_state
          gate = MCXGate(num_ctrl_qubits=num_ctrl_qubits + 3, label=label, ctrl_state=new_ctrl_state)
          gate.base_gate.label = self.label
          return gate
  
      def inverse(self):
          """Invert this gate. The C3X is its own inverse."""
          return C3XGate(angle=-self._angle, ctrl_state=self.ctrl_state)
  
      # This matrix is only correct if the angle is pi/4
      # def __array__(self, dtype=None):
      #     """Return a numpy.array for the C3X gate."""
      #     return numpy.array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
      #                         [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype)
  
  
  class RC3XGate(Gate):
      """The simplified 3-controlled Toffoli gate.
  
      The simplified Toffoli gate implements the Toffoli gate up to relative phases.
      Note, that the simplified Toffoli is not equivalent to the Toffoli. But can be used in places
      where the Toffoli gate is uncomputed again.
  
      This concrete implementation is from https://arxiv.org/abs/1508.03273, the complete circuit
      of Fig. 4.
      """
  
      def __init__(self, label=None):
          """Create a new RC3X gate."""
          super().__init__('rcccx', 4, [], label=label)
  
      def _define(self):
          """
          gate rc3x a,b,c,d
          { u2(0,pi) d;
            u1(pi/4) d;
            cx c,d;
            u1(-pi/4) d;
            u2(0,pi) d;
            cx a,d;
            u1(pi/4) d;
            cx b,d;
            u1(-pi/4) d;
            cx a,d;
            u1(pi/4) d;
            cx b,d;
            u1(-pi/4) d;
            u2(0,pi) d;
            u1(pi/4) d;
            cx c,d;
            u1(-pi/4) d;
            u2(0,pi) d;
          }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          q = QuantumRegister(4, 'q')
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (U2Gate(0, pi), [q[3]], []),  # H gate
              (U1Gate(pi / 4), [q[3]], []),  # T gate
              (CXGate(), [q[2], q[3]], []),
              (U1Gate(-pi / 4), [q[3]], []),  # inverse T gate
              (U2Gate(0, pi), [q[3]], []),
              (CXGate(), [q[0], q[3]], []),
              (U1Gate(pi / 4), [q[3]], []),
              (CXGate(), [q[1], q[3]], []),
              (U1Gate(-pi / 4), [q[3]], []),
              (CXGate(), [q[0], q[3]], []),
              (U1Gate(pi / 4), [q[3]], []),
              (CXGate(), [q[1], q[3]], []),
              (U1Gate(-pi / 4), [q[3]], []),
              (U2Gate(0, pi), [q[3]], []),
              (U1Gate(pi / 4), [q[3]], []),
              (CXGate(), [q[2], q[3]], []),
              (U1Gate(-pi / 4), [q[3]], []),
              (U2Gate(0, pi), [q[3]], []),
          ]
          for instr, qargs, cargs in rules:
              qc._append(instr, qargs, cargs)
  
          self.definition = qc
  
      def __array__(self, dtype=None):
          """Return a numpy.array for the RC3X gate."""
          return numpy.array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                              [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                              [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                              [0, 0, 0, 1j, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
                              [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1j, 0, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
                              [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
                              [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype)
  
  
  class C4XGate(ControlledGate):
      """The 4-qubit controlled X gate.
  
      This implementation is based on Page 21, Lemma 7.5, of [1], with the use
      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
      """
  
      def __init__(self, label=None, ctrl_state=None):
          """Create a new 4-qubit controlled X gate."""
          super().__init__('mcx', 5, [], num_ctrl_qubits=4, label=label,
                           ctrl_state=ctrl_state, base_gate=XGate())
  
      # seems like open controls not hapening?
      def _define(self):
          """
          gate c3sqrtx a,b,c,d
          {
              h d; cu1(-pi/8) a,d; h d;
              cx a,b;
              h d; cu1(pi/8) b,d; h d;
              cx a,b;
              h d; cu1(-pi/8) b,d; h d;
              cx b,c;
              h d; cu1(pi/8) c,d; h d;
              cx a,c;
              h d; cu1(-pi/8) c,d; h d;
              cx b,c;
              h d; cu1(pi/8) c,d; h d;
              cx a,c;
              h d; cu1(-pi/8) c,d; h d;
          }
          gate c4x a,b,c,d,e
          {
              h e; cu1(-pi/2) d,e; h e;
              rc3x a,b,c,d;
              h e; cu1(pi/4) d,e; h e;
              rc3x a,b,c,d;
              c3sqrtx a,b,c,e;
          }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .u1 import CU1Gate
          q = QuantumRegister(5, name='q')
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (HGate(), [q[4]], []),
              (CU1Gate(-numpy.pi / 2), [q[3], q[4]], []),
              (HGate(), [q[4]], []),
              (RC3XGate(), [q[0], q[1], q[2], q[3]], []),
              (HGate(), [q[4]], []),
              (CU1Gate(numpy.pi / 2), [q[3], q[4]], []),
              (HGate(), [q[4]], []),
              (RC3XGate().inverse(), [q[0], q[1], q[2], q[3]], []),
              (C3XGate(numpy.pi / 8), [q[0], q[1], q[2], q[4]], []),
          ]
          for instr, qargs, cargs in rules:
              qc._append(instr, qargs, cargs)
  
          self.definition = qc
  
      def control(self, num_ctrl_qubits=1, label=None, ctrl_state=None):
          """Controlled version of this gate.
  
          Args:
              num_ctrl_qubits (int): number of control qubits.
              label (str or None): An optional label for the gate [Default: None]
              ctrl_state (int or str or None): control state expressed as integer,
                  string (e.g. '110'), or None. If None, use all 1s.
  
          Returns:
              ControlledGate: controlled version of this gate.
          """
          ctrl_state = _ctrl_state_to_int(ctrl_state, num_ctrl_qubits)
          new_ctrl_state = (self.ctrl_state << num_ctrl_qubits) | ctrl_state
          gate = MCXGate(num_ctrl_qubits=num_ctrl_qubits + 4, label=label, ctrl_state=new_ctrl_state)
          gate.base_gate.label = self.label
          return gate
  
      def inverse(self):
          """Invert this gate. The C4X is its own inverse."""
          return C4XGate(ctrl_state=self.ctrl_state)
  
      def __array__(self, dtype=None):
          """Return a numpy.array for the C4X gate."""
          mat = _compute_control_matrix(self.base_gate.to_matrix(),
                                        self.num_ctrl_qubits,
                                        ctrl_state=self.ctrl_state)
          if dtype:
              return numpy.asarray(mat, dtype=dtype)
          return mat
  
  
  class MCXGate(ControlledGate):
      """The general, multi-controlled X gate."""
  
      def __new__(cls, num_ctrl_qubits=None, label=None, ctrl_state=None):
          """Create a new MCX instance.
  
          Depending on the number of controls, this creates an explicit CX, CCX, C3X or C4X
          instance or a generic MCX gate.
          """
          # these gates will always be implemented for all modes of the MCX if the number of control
          # qubits matches this
          explicit = {
              1: CXGate,
              2: CCXGate
          }
          if num_ctrl_qubits in explicit.keys():
              gate_class = explicit[num_ctrl_qubits]
              gate = gate_class.__new__(gate_class, label=label, ctrl_state=ctrl_state)
              # if __new__ does not return the same type as cls, init is not called
              gate.__init__(label=label, ctrl_state=ctrl_state)
              return gate
          return super().__new__(cls)
  
      def __init__(self, num_ctrl_qubits, label=None, ctrl_state=None, _name='mcx'):
          """Create new MCX gate."""
          num_ancilla_qubits = self.__class__.get_num_ancilla_qubits(num_ctrl_qubits)
          super().__init__(_name, num_ctrl_qubits + 1 + num_ancilla_qubits, [],
                           num_ctrl_qubits=num_ctrl_qubits, label=label,
                           ctrl_state=ctrl_state, base_gate=XGate())
  
      def inverse(self):
          """Invert this gate. The MCX is its own inverse."""
          return MCXGate(num_ctrl_qubits=self.num_ctrl_qubits, ctrl_state=self.ctrl_state)
  
      @staticmethod
      def get_num_ancilla_qubits(num_ctrl_qubits, mode='noancilla'):
          """Get the number of required ancilla qubits without instantiating the class.
  
          This staticmethod might be necessary to check the number of ancillas before
          creating the gate, or to use the number of ancillas in the initialization.
          """
          if mode == 'noancilla':
              return 0
          if mode in ['recursion', 'advanced']:
              return int(num_ctrl_qubits > 4)
          if mode[:7] == 'v-chain' or mode[:5] == 'basic':
              return max(0, num_ctrl_qubits - 2)
          raise AttributeError('Unsupported mode ({}) specified!'.format(mode))
  
      def _define(self):
          """The standard definition used the Gray code implementation."""
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          q = QuantumRegister(self.num_qubits, name='q')
          qc = QuantumCircuit(q)
          qc._append(MCXGrayCode(self.num_ctrl_qubits), q[:], [])
          self.definition = qc
  
      @property
      def num_ancilla_qubits(self):
          """The number of ancilla qubits."""
          return self.__class__.get_num_ancilla_qubits(self.num_ctrl_qubits)
  
      def control(self, num_ctrl_qubits=1, label=None, ctrl_state=None):
          """Return a multi-controlled-X gate with more control lines.
  
          Args:
              num_ctrl_qubits (int): number of control qubits.
              label (str or None): An optional label for the gate [Default: None]
              ctrl_state (int or str or None): control state expressed as integer,
                  string (e.g. '110'), or None. If None, use all 1s.
  
          Returns:
              ControlledGate: controlled version of this gate.
          """
          if ctrl_state is None:
              # use __class__ so this works for derived classes
              gate = self.__class__(self.num_ctrl_qubits + num_ctrl_qubits, label=label,
                                    ctrl_state=ctrl_state)
              gate.base_gate.label = self.label
              return gate
          return super().control(num_ctrl_qubits, label=label, ctrl_state=ctrl_state)
  
  
  class MCXGrayCode(MCXGate):
      r"""Implement the multi-controlled X gate using the Gray code.
  
      This delegates the implementation to the MCU1 gate, since :math:`X = H \cdot U1(\pi) \cdot H`.
      """
  
      def __init__(self, num_ctrl_qubits, label=None, ctrl_state=None):
          super().__init__(num_ctrl_qubits, label=label, ctrl_state=ctrl_state, _name='mcx_gray')
  
      def inverse(self):
          """Invert this gate. The MCX is its own inverse."""
          return MCXGrayCode(num_ctrl_qubits=self.num_ctrl_qubits, ctrl_state=self.ctrl_state)
  
      def _define(self):
          """Define the MCX gate using the Gray code."""
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .u1 import MCU1Gate
          q = QuantumRegister(self.num_qubits, name='q')
          qc = QuantumCircuit(q, name=self.name)
          qc._append(HGate(), [q[-1]], [])
          qc._append(MCU1Gate(numpy.pi, num_ctrl_qubits=self.num_ctrl_qubits), q[:], [])
          qc._append(HGate(), [q[-1]], [])
          self.definition = qc
  
  
  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.
      """
  
      def __init__(self, num_ctrl_qubits, label=None, ctrl_state=None):
          super().__init__(num_ctrl_qubits, label=label, ctrl_state=ctrl_state, _name='mcx_recursive')
  
      @staticmethod
      def get_num_ancilla_qubits(num_ctrl_qubits, mode='recursion'):
          """Get the number of required ancilla qubits."""
          return MCXGate.get_num_ancilla_qubits(num_ctrl_qubits, mode)
  
      def inverse(self):
          """Invert this gate. The MCX is its own inverse."""
          return MCXRecursive(num_ctrl_qubits=self.num_ctrl_qubits, ctrl_state=self.ctrl_state)
  
      def _define(self):
          """Define the MCX gate using recursion."""
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          q = QuantumRegister(self.num_qubits, name='q')
          qc = QuantumCircuit(q, name=self.name)
          if self.num_qubits == 4:
              qc._append(C3XGate(), q[:], [])
              self.definition = qc
          elif self.num_qubits == 5:
              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)
              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."""
  
      def __new__(cls, num_ctrl_qubits=None, dirty_ancillas=False,  # pylint: disable=unused-argument
                  label=None, ctrl_state=None):
          """Create a new MCX instance.
  
          This must be defined anew to include the additional argument ``dirty_ancillas``.
          """
          return super().__new__(cls, num_ctrl_qubits, label=label, ctrl_state=ctrl_state)
  
      def __init__(self, num_ctrl_qubits, dirty_ancillas=False, label=None, ctrl_state=None):
          super().__init__(num_ctrl_qubits, label=label, ctrl_state=ctrl_state, _name='mcx_vchain')
          self._dirty_ancillas = dirty_ancillas
  
      def inverse(self):
          """Invert this gate. The MCX is its own inverse."""
          return MCXVChain(num_ctrl_qubits=self.num_ctrl_qubits,
                           dirty_ancillas=self._dirty_ancillas,
                           ctrl_state=self.ctrl_state)
  
      @staticmethod
      def get_num_ancilla_qubits(num_ctrl_qubits, mode='v-chain'):
          """Get the number of required ancilla qubits."""
          return MCXGate.get_num_ancilla_qubits(num_ctrl_qubits, mode)
  
      def _define(self):
          """Define the MCX gate using a V-chain of CX gates."""
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          q = QuantumRegister(self.num_qubits, name='q')
          qc = QuantumCircuit(q, name=self.name)
          q_controls = q[:self.num_ctrl_qubits]
          q_target = q[self.num_ctrl_qubits]
          q_ancillas = q[self.num_ctrl_qubits + 1:]
  
          definition = []
  
          if self._dirty_ancillas:
              i = self.num_ctrl_qubits - 3
              ancilla_pre_rule = [
                  (U2Gate(0, numpy.pi), [q_target], []),
                  (CXGate(), [q_target, q_ancillas[i]], []),
                  (U1Gate(-numpy.pi / 4), [q_ancillas[i]], []),
                  (CXGate(), [q_controls[-1], q_ancillas[i]], []),
                  (U1Gate(numpy.pi / 4), [q_ancillas[i]], []),
                  (CXGate(), [q_target, q_ancillas[i]], []),
                  (U1Gate(-numpy.pi / 4), [q_ancillas[i]], []),
                  (CXGate(), [q_controls[-1], q_ancillas[i]], []),
                  (U1Gate(numpy.pi / 4), [q_ancillas[i]], []),
              ]
              for inst in ancilla_pre_rule:
                  definition.append(inst)
  
              for j in reversed(range(2, self.num_ctrl_qubits - 1)):
                  definition.append(
                      (RCCXGate(), [q_controls[j], q_ancillas[i - 1], q_ancillas[i]], []))
                  i -= 1
  
          definition.append((RCCXGate(), [q_controls[0], q_controls[1], q_ancillas[0]], []))
          i = 0
          for j in range(2, self.num_ctrl_qubits - 1):
              definition.append((RCCXGate(), [q_controls[j], q_ancillas[i], q_ancillas[i + 1]], []))
              i += 1
  
          if self._dirty_ancillas:
              ancilla_post_rule = [
                  (U1Gate(-numpy.pi / 4), [q_ancillas[i]], []),
                  (CXGate(), [q_controls[-1], q_ancillas[i]], []),
                  (U1Gate(numpy.pi / 4), [q_ancillas[i]], []),
                  (CXGate(), [q_target, q_ancillas[i]], []),
                  (U1Gate(-numpy.pi / 4), [q_ancillas[i]], []),
                  (CXGate(), [q_controls[-1], q_ancillas[i]], []),
                  (U1Gate(numpy.pi / 4), [q_ancillas[i]], []),
                  (CXGate(), [q_target, q_ancillas[i]], []),
                  (U2Gate(0, numpy.pi), [q_target], []),
              ]
              for inst in ancilla_post_rule:
                  definition.append(inst)
          else:
              definition.append((CCXGate(), [q_controls[-1], q_ancillas[i], q_target], []))
  
          for j in reversed(range(2, self.num_ctrl_qubits - 1)):
              definition.append((RCCXGate(), [q_controls[j], q_ancillas[i - 1], q_ancillas[i]], []))
              i -= 1
          definition.append((RCCXGate(), [q_controls[0], q_controls[1], q_ancillas[i]], []))
  
          if self._dirty_ancillas:
              for i, j in enumerate(list(range(2, self.num_ctrl_qubits - 1))):
                  definition.append(
                      (RCCXGate(), [q_controls[j], q_ancillas[i], q_ancillas[i + 1]], []))
  
          for instr, qargs, cargs in definition:
              qc._append(instr, qargs, cargs)
          self.definition = qc
  