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rzz.py
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rzz.py
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# 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.
"""Two-qubit ZZ-rotation gate."""
from cmath import exp
from typing import Optional
from qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister
from qiskit.circuit.parameterexpression import ParameterValueType
class RZZGate(Gate):
r"""A parametric 2-qubit :math:`Z \otimes Z` interaction (rotation about ZZ).
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.rzz` method.
**Circuit Symbol:**
.. parsed-literal::
q_0: ───■────
│zz(θ)
q_1: ───■────
**Matrix Representation:**
.. math::
\newcommand{\rotationangle}{\frac{\theta}{2}}
R_{ZZ}(\theta) = \exp\left(-i \rotationangle Z{\otimes}Z\right) =
\begin{pmatrix}
e^{-i \rotationangle} & 0 & 0 & 0 \\
0 & e^{i \rotationangle} & 0 & 0 \\
0 & 0 & e^{i \rotationangle} & 0 \\
0 & 0 & 0 & e^{-i \rotationangle}
\end{pmatrix}
This is a direct sum of RZ rotations, so this gate is equivalent to a
uniformly controlled (multiplexed) RZ gate:
.. math::
R_{ZZ}(\theta) =
\begin{pmatrix}
RZ(\theta) & 0 \\
0 & RZ(-\theta)
\end{pmatrix}
**Examples:**
.. math::
R_{ZZ}(\theta = 0) = I
.. math::
R_{ZZ}(\theta = 2\pi) = -I
.. math::
R_{ZZ}(\theta = \pi) = - Z \otimes Z
.. math::
R_{ZZ}\left(\theta = \frac{\pi}{2}\right) = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1-i & 0 & 0 & 0 \\
0 & 1+i & 0 & 0 \\
0 & 0 & 1+i & 0 \\
0 & 0 & 0 & 1-i
\end{pmatrix}
"""
def __init__(
self, theta: ParameterValueType, label: Optional[str] = None, *, duration=None, unit="dt"
):
"""Create new RZZ gate."""
super().__init__("rzz", 2, [theta], label=label, duration=duration, unit=unit)
def _define(self):
"""
gate rzz(theta) a, b { cx a, b; u1(theta) b; cx a, b; }
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .x import CXGate
from .rz import RZGate
# q_0: ──■─────────────■──
# ┌─┴─┐┌───────┐┌─┴─┐
# q_1: ┤ X ├┤ Rz(0) ├┤ X ├
# └───┘└───────┘└───┘
q = QuantumRegister(2, "q")
theta = self.params[0]
qc = QuantumCircuit(q, name=self.name)
rules = [
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
def inverse(self, annotated: bool = False):
"""Return inverse RZZ gate (i.e. with the negative rotation angle).
Args:
annotated: when set to ``True``, this is typically used to return an
:class:`.AnnotatedOperation` with an inverse modifier set instead of a concrete
:class:`.Gate`. However, for this class this argument is ignored as the inverse
of this gate is always a :class:`.RZZGate` with an inverted parameter value.
Returns:
RZZGate: inverse gate.
"""
return RZZGate(-self.params[0])
def __array__(self, dtype=None, copy=None):
"""Return a numpy.array for the RZZ gate."""
import numpy
if copy is False:
raise ValueError("unable to avoid copy while creating an array as requested")
itheta2 = 1j * float(self.params[0]) / 2
return numpy.array(
[
[exp(-itheta2), 0, 0, 0],
[0, exp(itheta2), 0, 0],
[0, 0, exp(itheta2), 0],
[0, 0, 0, exp(-itheta2)],
],
dtype=dtype,
)
def power(self, exponent: float, annotated: bool = False):
(theta,) = self.params
return RZZGate(exponent * theta)
def __eq__(self, other):
if isinstance(other, RZZGate):
return self._compare_parameters(other)
return False