# This code is part of Qiskit.
#
# (C) Copyright IBM 2021.
#
# 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-YY gate."""
import math
from cmath import exp
from math import pi
from typing import Optional
import numpy as np
from qiskit.circuit.gate import Gate
from qiskit.circuit.library.standard_gates.ry import RYGate
from qiskit.circuit.library.standard_gates.rz import RZGate
from qiskit.circuit.library.standard_gates.s import SdgGate, SGate
from qiskit.circuit.library.standard_gates.sx import SXdgGate, SXGate
from qiskit.circuit.library.standard_gates.x import CXGate
from qiskit.circuit.parameterexpression import ParameterValueType
from qiskit.circuit.quantumcircuit import QuantumCircuit
from qiskit.circuit.quantumregister import QuantumRegister
[docs]class XXMinusYYGate(Gate):
r"""XX-YY interaction gate.
A 2-qubit parameterized XX-YY interaction. Its action is to induce
a coherent rotation by some angle between :math:`|00\rangle` and :math:`|11\rangle`.
**Circuit Symbol:**
.. parsed-literal::
┌───────────────┐
q_0: ┤0 ├
│ (XX-YY)(θ,β) │
q_1: ┤1 ├
└───────────────┘
**Matrix Representation:**
.. math::
\newcommand{\th}{\frac{\theta}{2}}
R_{XX-YY}(\theta, \beta) q_0, q_1 =
RZ_1(\beta) \cdot \exp\left(-i \frac{\theta}{2} \frac{XX-YY}{2}\right) \cdot RZ_1(-\beta) =
\begin{pmatrix}
\cos\left(\th\right) & 0 & 0 & -i\sin\left(\th\right)e^{-i\beta} \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-i\sin\left(\th\right)e^{i\beta} & 0 & 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 adding the (optional) phase defined
by :math:`beta` on q_1. Instead, if we apply it on (q_1, q_0), the
phase is added on q_0. If :math:`beta` is set to its default value
of :math:`0`, the gate is equivalent in big and little endian.
.. parsed-literal::
┌───────────────┐
q_0: ┤1 ├
│ (XX-YY)(θ,β) │
q_1: ┤0 ├
└───────────────┘
.. math::
\newcommand{\th}{\frac{\theta}{2}}
R_{XX-YY}(\theta, \beta) q_1, q_0 =
RZ_0(\beta) \cdot \exp\left(-i \frac{\theta}{2} \frac{XX-YY}{2}\right) \cdot RZ_0(-\beta) =
\begin{pmatrix}
\cos\left(\th\right) & 0 & 0 & -i\sin\left(\th\right)e^{i\beta} \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-i\sin\left(\th\right)e^{-i\beta} & 0 & 0 & \cos\left(\th\right)
\end{pmatrix}
"""
def __init__(
self,
theta: ParameterValueType,
beta: ParameterValueType = 0,
label: Optional[str] = "(XX-YY)",
):
"""Create new XX-YY gate.
Args:
theta: The rotation angle.
beta: The phase angle.
label: The label of the gate.
"""
super().__init__("xx_minus_yy", 2, [theta, beta], label=label)
def _define(self):
"""
gate xx_minus_yy(theta, beta) a, b {
rz(-beta) b;
rz(-pi/2) a;
sx a;
rz(pi/2) a;
s b;
cx a, b;
ry(theta/2) a;
ry(-theta/2) b;
cx a, b;
sdg b;
rz(-pi/2) a;
sxdg a;
rz(pi/2) a;
rz(beta) b;
}
"""
theta, beta = self.params
register = QuantumRegister(2, "q")
circuit = QuantumCircuit(register, name=self.name)
a, b = register
rules = [
(RZGate(-beta), [b], []),
(RZGate(-pi / 2), [a], []),
(SXGate(), [a], []),
(RZGate(pi / 2), [a], []),
(SGate(), [b], []),
(CXGate(), [a, b], []),
(RYGate(theta / 2), [a], []),
(RYGate(-theta / 2), [b], []),
(CXGate(), [a, b], []),
(SdgGate(), [b], []),
(RZGate(-pi / 2), [a], []),
(SXdgGate(), [a], []),
(RZGate(pi / 2), [a], []),
(RZGate(beta), [b], []),
]
for instr, qargs, cargs in rules:
circuit._append(instr, qargs, cargs)
self.definition = circuit
[docs] def inverse(self):
"""Inverse gate."""
theta, beta = self.params
return XXMinusYYGate(-theta, beta)
def __array__(self, dtype=complex):
"""Gate matrix."""
theta, beta = self.params
cos = math.cos(theta / 2)
sin = math.sin(theta / 2)
return np.array(
[
[cos, 0, 0, -1j * sin * exp(-1j * beta)],
[0, 1, 0, 0],
[0, 0, 1, 0],
[-1j * sin * exp(1j * beta), 0, 0, cos],
],
dtype=dtype,
)
[docs] def power(self, exponent: float):
"""Raise gate to a power."""
theta, beta = self.params
return XXMinusYYGate(exponent * theta, beta)