# 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.

"""One-pulse single-qubit gate."""
from math import sqrt, pi
from cmath import exp
from typing import Optional
import numpy
from qiskit.circuit.gate import Gate
from qiskit.circuit.parameterexpression import ParameterValueType
from qiskit.circuit.quantumregister import QuantumRegister


class U2Gate(Gate):
    r"""Single-qubit rotation about the X+Z axis.

    Implemented using one X90 pulse on IBM Quantum systems:

    .. warning::

       This gate is deprecated. Instead, the following replacements should be used

       .. math::

           U2(\phi, \lambda) = U\left(\frac{\pi}{2}, \phi, \lambda\right)

       .. code-block:: python

          circuit = QuantumCircuit(1)
          circuit.u(pi/2, phi, lambda)



    **Circuit symbol:**

    .. parsed-literal::

             ┌─────────┐
        q_0: ┤ U2(φ,λ) ├
             └─────────┘

    **Matrix Representation:**

    .. math::

        U2(\phi, \lambda) = \frac{1}{\sqrt{2}}
            \begin{pmatrix}
                1          & -e^{i\lambda} \\
                e^{i\phi} & e^{i(\phi+\lambda)}
            \end{pmatrix}

    **Examples:**

    .. math::

        U2(\phi,\lambda) = e^{i \frac{\phi + \lambda}{2}}RZ(\phi)
        RY\left(\frac{\pi}{2}\right) RZ(\lambda)
        = e^{- i\frac{\pi}{4}} P\left(\frac{\pi}{2} + \phi\right)
        \sqrt{X} P\left(\lambda- \frac{\pi}{2}\right)

    .. math::

        U2(0, \pi) = H

    .. math::

        U2(0, 0) = RY(\pi/2)

    .. math::

        U2(-\pi/2, \pi/2) = RX(\pi/2)

    .. seealso::

        :class:`~qiskit.circuit.library.standard_gates.U3Gate`:
        U3 is a generalization of U2 that covers all single-qubit rotations,
        using two X90 pulses.
    """

    def __init__(
        self, phi: ParameterValueType, lam: ParameterValueType, label: Optional[str] = None
    ):
        """Create new U2 gate."""
        super().__init__("u2", 1, [phi, lam], label=label)

    def _define(self):
        # 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 / 2, self.params[0], self.params[1]), [q[0]], [])]
        for instr, qargs, cargs in rules:
            qc._append(instr, qargs, cargs)

        self.definition = qc

    def inverse(self):
        r"""Return inverted U2 gate.

        :math:`U2(\phi, \lambda)^{\dagger} =U2(-\lambda-\pi, -\phi+\pi)`)
        """
        return U2Gate(-self.params[1] - pi, -self.params[0] + pi)

    def __array__(self, dtype=complex):
        """Return a Numpy.array for the U2 gate."""
        isqrt2 = 1 / sqrt(2)
        phi, lam = self.params
        phi, lam = float(phi), float(lam)
        return numpy.array(
            [
                [isqrt2, -exp(1j * lam) * isqrt2],
                [exp(1j * phi) * isqrt2, exp(1j * (phi + lam)) * isqrt2],
            ],
            dtype=dtype,
        )
