# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2019.
#
# 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.
"""Two-qubit XX-rotation gate."""
import math
from typing import Optional
import numpy
from qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister
from qiskit.circuit.parameterexpression import ParameterValueType
[docs]class RXXGate(Gate):
r"""A parametric 2-qubit :math:`X \otimes X` interaction (rotation about XX).
This gate is symmetric, and is maximally entangling at :math:`\theta = \pi/2`.
Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
with the :meth:`~qiskit.circuit.QuantumCircuit.rxx` method.
**Circuit Symbol:**
.. parsed-literal::
┌─────────┐
q_0: ┤1 ├
│ Rxx(ϴ) │
q_1: ┤0 ├
└─────────┘
**Matrix Representation:**
.. math::
\newcommand{\th}{\frac{\theta}{2}}
R_{XX}(\theta) = \exp\left(-i \th X{\otimes}X\right) =
\begin{pmatrix}
\cos\left(\th\right) & 0 & 0 & -i\sin\left(\th\right) \\
0 & \cos\left(\th\right) & -i\sin\left(\th\right) & 0 \\
0 & -i\sin\left(\th\right) & \cos\left(\th\right) & 0 \\
-i\sin\left(\th\right) & 0 & 0 & \cos\left(\th\right)
\end{pmatrix}
**Examples:**
.. math::
R_{XX}(\theta = 0) = I
.. math::
R_{XX}(\theta = \pi) = i X \otimes X
.. math::
R_{XX}\left(\theta = \frac{\pi}{2}\right) = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 0 & 0 & -i \\
0 & 1 & -i & 0 \\
0 & -i & 1 & 0 \\
-i & 0 & 0 & 1
\end{pmatrix}
"""
def __init__(self, theta: ParameterValueType, label: Optional[str] = None):
"""Create new RXX gate."""
super().__init__("rxx", 2, [theta], label=label)
def _define(self):
"""Calculate a subcircuit that implements this unitary."""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .x import CXGate
from .h import HGate
from .rz import RZGate
# ┌───┐ ┌───┐
# q_0: ┤ H ├──■─────────────■──┤ H ├
# ├───┤┌─┴─┐┌───────┐┌─┴─┐├───┤
# q_1: ┤ H ├┤ X ├┤ Rz(0) ├┤ X ├┤ H ├
# └───┘└───┘└───────┘└───┘└───┘
theta = self.params[0]
q = QuantumRegister(2, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [
(HGate(), [q[0]], []),
(HGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(HGate(), [q[1]], []),
(HGate(), [q[0]], []),
]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
[docs] def inverse(self):
"""Return inverse RXX gate (i.e. with the negative rotation angle)."""
return RXXGate(-self.params[0])
def __array__(self, dtype=None):
"""Return a Numpy.array for the RXX gate."""
theta2 = float(self.params[0]) / 2
cos = math.cos(theta2)
isin = 1j * math.sin(theta2)
return numpy.array(
[[cos, 0, 0, -isin], [0, cos, -isin, 0], [0, -isin, cos, 0], [-isin, 0, 0, cos]],
dtype=dtype,
)
[docs] def power(self, exponent: float):
"""Raise gate to a power."""
(theta,) = self.params
return RXXGate(exponent * theta)