code: # 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.
  
  """The S and Sdg gate."""
  
  import numpy
  from qiskit.qasm import pi
  from qiskit.circuit.gate import Gate
  from qiskit.circuit.quantumregister import QuantumRegister
  
  
  class SGate(Gate):
      r"""Single qubit S gate (Z**0.5).
  
      It induces a :math:`\pi/2` phase, and is sometimes called the P gate (phase).
  
      This is a Clifford gate and a square-root of Pauli-Z.
  
      **Matrix Representation:**
  
      .. math::
  
          S = \begin{pmatrix}
                  1 & 0 \\
                  0 & i
              \end{pmatrix}
  
      **Circuit symbol:**
  
      .. parsed-literal::
  
               ┌───┐
          q_0: ┤ S ├
               └───┘
  
      Equivalent to a :math:`\pi/2` radian rotation about the Z axis.
      """
  
      def __init__(self, label=None):
          """Create new S gate."""
          super().__init__('s', 1, [], label=label)
  
      def _define(self):
          """
          gate s a { u1(pi/2) a; }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .u1 import U1Gate
          q = QuantumRegister(1, 'q')
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (U1Gate(pi / 2), [q[0]], [])
          ]
          qc._data = rules
          self.definition = qc
  
      def inverse(self):
          """Return inverse of S (SdgGate)."""
          return SdgGate()
  
      def to_matrix(self):
          """Return a numpy.array for the S gate."""
          return numpy.array([[1, 0],
                              [0, 1j]], dtype=complex)
  
  
  class SdgGate(Gate):
      r"""Single qubit S-adjoint gate (~Z**0.5).
  
      It induces a :math:`-\pi/2` phase.
  
      This is a Clifford gate and a square-root of Pauli-Z.
  
      **Matrix Representation:**
  
      .. math::
  
          Sdg = \begin{pmatrix}
                  1 & 0 \\
                  0 & -i
              \end{pmatrix}
  
      **Circuit symbol:**
  
      .. parsed-literal::
  
               ┌─────┐
          q_0: ┤ Sdg ├
               └─────┘
  
      Equivalent to a :math:`\pi/2` radian rotation about the Z axis.
      """
  
      def __init__(self, label=None):
          """Create new Sdg gate."""
          super().__init__('sdg', 1, [], label=label)
  
      def _define(self):
          """
          gate sdg a { u1(-pi/2) a; }
          """
          # pylint: disable=cyclic-import
          from qiskit.circuit.quantumcircuit import QuantumCircuit
          from .u1 import U1Gate
          q = QuantumRegister(1, 'q')
          qc = QuantumCircuit(q, name=self.name)
          rules = [
              (U1Gate(-pi / 2), [q[0]], [])
          ]
          qc._data = rules
          self.definition = qc
  
      def inverse(self):
          """Return inverse of Sdg (SGate)."""
          return SGate()
  
      def to_matrix(self):
          """Return a numpy.array for the Sdg gate."""
          return numpy.array([[1, 0],
                              [0, -1j]], dtype=complex)
  
apis: numpy.array