# 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
[docs]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
[docs] 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,
)