# -*- coding: utf-8 -*-
# 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."""
import numpy
from qiskit.qasm import pi
from qiskit.circuit.gate import Gate
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:
.. math::
U2(\phi, \lambda) = RZ(\phi+\pi/2).RX(\frac{\pi}{2}).RZ(\lambda-\pi/2)
**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(0, \pi) = H
.. 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, lam, label=None):
"""Create new U2 gate."""
super().__init__('u2', 1, [phi, lam], label=label)
def _define(self):
from .u3 import U3Gate
definition = []
q = QuantumRegister(1, 'q')
rule = [(U3Gate(pi / 2, self.params[0], self.params[1]), [q[0]], [])]
for inst in rule:
definition.append(inst)
self.definition = definition
[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)
[docs] def to_matrix(self):
"""Return a Numpy.array for the U2 gate."""
isqrt2 = 1 / numpy.sqrt(2)
phi, lam = self.params
phi, lam = float(phi), float(lam)
return numpy.array([
[
isqrt2,
-numpy.exp(1j * lam) * isqrt2
],
[
numpy.exp(1j * phi) * isqrt2,
numpy.exp(1j * (phi + lam)) * isqrt2
]
], dtype=complex)