# Source code for qiskit.circuit.library.standard_gates.rzx

# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020.
#
# obtain a copy of this license in the LICENSE.txt file in the root directory
#
# 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

[docs]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

[docs]    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,
)

[docs]    def power(self, exponent: float):
"""Raise gate to a power."""
(theta,) = self.params
return RZXGate(exponent * theta)