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.
  
  """Hadamard gate."""
  
  import numpy
  from qiskit.circuit.controlledgate import ControlledGate
  from qiskit.circuit.gate import Gate
  from qiskit.circuit.quantumregister import QuantumRegister
  from qiskit.qasm import pi
  from .t import TGate, TdgGate
  from .s import SGate, SdgGate
  
  
  class HGate(Gate):
      r"""Single-qubit Hadamard gate.
  
      This gate is a \pi rotation about the X+Z axis, and has the effect of
      changing computation basis from :math:`|0\rangle,|1\rangle` to
      :math:`|+\rangle,|-\rangle` and vice-versa.
  
      **Circuit symbol:**
  
      .. parsed-literal::
  
               ┌───┐
          q_0: ┤ H ├
               └───┘
  
      **Matrix Representation:**
  
      .. math::
  
          H = \frac{1}{\sqrt{2}}
              \begin{pmatrix}
                  1 & 1 \\
                  1 & -1
              \end{pmatrix}
      """
  
      def __init__(self, label=None):
          """Create new H gate."""
          super().__init__("h", 1, [], label=label)
  
      def _define(self):
          """
          gate h a { u2(0,pi) a; }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .u2 import U2Gate
  
          q = QuantumRegister(1, "q")
          qc = QuantumCircuit(q, name=self.name)
          rules = [(U2Gate(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 (multi-)controlled-H gate.
  
          One control qubit returns a CH 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.
          """
          if num_ctrl_qubits == 1:
              gate = CHGate(label=label, ctrl_state=ctrl_state)
              gate.base_gate.label = self.label
              return gate
          return super().control(num_ctrl_qubits=num_ctrl_qubits, label=label, ctrl_state=ctrl_state)
  
      def inverse(self):
          r"""Return inverted H gate (itself)."""
          return HGate()  # self-inverse
  
      def __array__(self, dtype=None):
          """Return a Numpy.array for the H gate."""
          return numpy.array([[1, 1], [1, -1]], dtype=dtype) / numpy.sqrt(2)
  
  
  class CHGate(ControlledGate):
      r"""Controlled-Hadamard gate.
  
      Applies a Hadamard on the target qubit if the control is
      in the :math:`|1\rangle` state.
  
      **Circuit symbol:**
  
      .. parsed-literal::
  
          q_0: ──■──
               ┌─┴─┐
          q_1: ┤ H ├
               └───┘
  
      **Matrix Representation:**
  
      .. math::
  
          CH\ q_0, q_1 =
              I \otimes |0\rangle\langle 0| + H \otimes |1\rangle\langle 1| =
              \begin{pmatrix}
                  1 & 0 & 0 & 0 \\
                  0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
                  0 & 0 & 1 & 0 \\
                  0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
              \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: ┤ H ├
                   └─┬─┘
              q_1: ──■──
  
          .. math::
  
              CH\ q_1, q_0 =
                  |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes H =
                  \begin{pmatrix}
                      1 & 0 & 0 & 0 \\
                      0 & 1 & 0 & 0 \\
                      0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
                      0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
                  \end{pmatrix}
      """
      # Define class constants. This saves future allocation time.
      _sqrt2o2 = 1 / numpy.sqrt(2)
      _matrix1 = numpy.array(
          [[1, 0, 0, 0], [0, _sqrt2o2, 0, _sqrt2o2], [0, 0, 1, 0], [0, _sqrt2o2, 0, -_sqrt2o2]],
          dtype=complex,
      )
      _matrix0 = numpy.array(
          [[_sqrt2o2, 0, _sqrt2o2, 0], [0, 1, 0, 0], [_sqrt2o2, 0, -_sqrt2o2, 0], [0, 0, 0, 1]],
          dtype=complex,
      )
  
      def __init__(self, label=None, ctrl_state=None):
          """Create new CH gate."""
          super().__init__(
              "ch", 2, [], num_ctrl_qubits=1, label=label, ctrl_state=ctrl_state, base_gate=HGate()
          )
  
      def _define(self):
          """
          gate ch a,b {
              s b;
              h b;
              t b;
              cx a, b;
              tdg b;
              h b;
              sdg b;
          }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .x import CXGate  # pylint: disable=cyclic-import
  
          q = QuantumRegister(2, "q")
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (SGate(), [q[1]], []),
              (HGate(), [q[1]], []),
              (TGate(), [q[1]], []),
              (CXGate(), [q[0], q[1]], []),
              (TdgGate(), [q[1]], []),
              (HGate(), [q[1]], []),
              (SdgGate(), [q[1]], []),
          ]
          for instr, qargs, cargs in rules:
              qc._append(instr, qargs, cargs)
  
          self.definition = qc
  
      def inverse(self):
          """Return inverted CH gate (itself)."""
          return CHGate(ctrl_state=self.ctrl_state)  # self-inverse
  
      def __array__(self, dtype=None):
          """Return a numpy.array for the CH gate."""
          mat = self._matrix1 if self.ctrl_state else self._matrix0
          if dtype:
              return numpy.asarray(mat, dtype=dtype)
          return mat
  