# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020.
#
# 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 ZX-rotation gate."""
import math
from typing import Optional
from qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister
from qiskit.circuit.parameterexpression import ParameterValueType
[ドキュメント]class RZXGate(Gate):
r"""A parametric 2-qubit :math:`Z \otimes X` interaction (rotation about ZX).
This gate is maximally entangling at :math:`\theta = \pi/2`.
The cross-resonance gate (CR) for superconducting qubits implements
a ZX interaction (however other terms are also present in an experiment).
Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
with the :meth:`~qiskit.circuit.QuantumCircuit.rzx` method.
**Circuit Symbol:**
.. parsed-literal::
┌─────────┐
q_0: ┤0 ├
│ Rzx(θ) │
q_1: ┤1 ├
└─────────┘
**Matrix Representation:**
.. math::
\newcommand{\th}{\frac{\theta}{2}}
R_{ZX}(\theta)\ q_0, q_1 = \exp\left(-i \frac{\theta}{2} X{\otimes}Z\right) =
\begin{pmatrix}
\cos\left(\th\right) & 0 & -i\sin\left(\th\right) & 0 \\
0 & \cos\left(\th\right) & 0 & i\sin\left(\th\right) \\
-i\sin\left(\th\right) & 0 & \cos\left(\th\right) & 0 \\
0 & i\sin\left(\th\right) & 0 & \cos\left(\th\right)
\end{pmatrix}
.. note::
In Qiskit's convention, higher qubit indices are more significant
(little endian convention). In the above example we apply the gate
on (q_0, q_1) which results in the :math:`X \otimes Z` tensor order.
Instead, if we apply it on (q_1, q_0), the matrix will
be :math:`Z \otimes X`:
.. parsed-literal::
┌─────────┐
q_0: ┤1 ├
│ Rzx(θ) │
q_1: ┤0 ├
└─────────┘
.. math::
\newcommand{\th}{\frac{\theta}{2}}
R_{ZX}(\theta)\ q_1, q_0 = exp(-i \frac{\theta}{2} Z{\otimes}X) =
\begin{pmatrix}
\cos(\th) & -i\sin(\th) & 0 & 0 \\
-i\sin(\th) & \cos(\th) & 0 & 0 \\
0 & 0 & \cos(\th) & i\sin(\th) \\
0 & 0 & i\sin(\th) & \cos(\th)
\end{pmatrix}
This is a direct sum of RX rotations, so this gate is equivalent to a
uniformly controlled (multiplexed) RX gate:
.. math::
R_{ZX}(\theta)\ q_1, q_0 =
\begin{pmatrix}
RX(\theta) & 0 \\
0 & RX(-\theta)
\end{pmatrix}
**Examples:**
.. math::
R_{ZX}(\theta = 0) = I
.. math::
R_{ZX}(\theta = 2\pi) = -I
.. math::
R_{ZX}(\theta = \pi) = -i Z \otimes X
.. math::
RZX(\theta = \frac{\pi}{2}) = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 0 & -i & 0 \\
0 & 1 & 0 & i \\
-i & 0 & 1 & 0 \\
0 & i & 0 & 1
\end{pmatrix}
"""
def __init__(self, theta: ParameterValueType, label: Optional[str] = None):
"""Create new RZX gate."""
super().__init__("rzx", 2, [theta], label=label)
def _define(self):
"""
gate rzx(theta) a, b { h b; cx a, b; u1(theta) b; cx a, b; h b;}
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .h import HGate
from .x import CXGate
from .rz import RZGate
# q_0: ───────■─────────────■───────
# ┌───┐┌─┴─┐┌───────┐┌─┴─┐┌───┐
# 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[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(HGate(), [q[1]], []),
]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
[ドキュメント] def inverse(self):
"""Return inverse RZX gate (i.e. with the negative rotation angle)."""
return RZXGate(-self.params[0])
def __array__(self, dtype=None):
"""Return a numpy.array for the RZX gate."""
import numpy
half_theta = float(self.params[0]) / 2
cos = math.cos(half_theta)
isin = 1j * math.sin(half_theta)
return numpy.array(
[[cos, 0, -isin, 0], [0, cos, 0, isin], [-isin, 0, cos, 0], [0, isin, 0, cos]],
dtype=dtype,
)
[ドキュメント] def power(self, exponent: float):
"""Raise gate to a power."""
(theta,) = self.params
return RZXGate(exponent * theta)