# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 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 ZX-rotation gate."""
from math import sqrt
import numpy as np
from qiskit.circuit._utils import with_gate_array
from qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister
from .rzx import RZXGate
from .x import XGate
[docs]@with_gate_array(
sqrt(0.5) * np.array([[0, 1, 0, 1.0j], [1, 0, -1.0j, 0], [0, 1.0j, 0, 1], [-1.0j, 0, 1, 0]])
)
class ECRGate(Gate):
r"""An echoed cross-resonance gate.
This gate is maximally entangling and is equivalent to a CNOT up to
single-qubit pre-rotations. The echoing procedure mitigates some
unwanted terms (terms other than ZX) to cancel in an experiment.
More specifically, this gate implements :math:`\frac{1}{\sqrt{2}}(IX-XY)`.
Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
with the :meth:`~qiskit.circuit.QuantumCircuit.ecr` method.
**Circuit Symbol:**
.. parsed-literal::
┌─────────┐ ┌────────────┐┌────────┐┌─────────────┐
q_0: ┤0 ├ q_0: ┤0 ├┤ RX(pi) ├┤0 ├
│ ECR │ = │ RZX(pi/4) │└────────┘│ RZX(-pi/4) │
q_1: ┤1 ├ q_1: ┤1 ├──────────┤1 ├
└─────────┘ └────────────┘ └─────────────┘
**Matrix Representation:**
.. math::
ECR\ q_0, q_1 = \frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 1 & 0 & i \\
1 & 0 & -i & 0 \\
0 & i & 0 & 1 \\
-i & 0 & 1 & 0
\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 ├
│ ECR │
q_1: ┤0 ├
└─────────┘
.. math::
ECR\ q_0, q_1 = \frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 0 & 1 & i \\
0 & 0 & i & 1 \\
1 & -i & 0 & 0 \\
-i & 1 & 0 & 0
\end{pmatrix}
"""
def __init__(self):
"""Create new ECR gate."""
super().__init__("ecr", 2, [])
def _define(self):
"""
gate ecr a, b { rzx(pi/4) a, b; x a; rzx(-pi/4) a, b;}
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
q = QuantumRegister(2, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [
(RZXGate(np.pi / 4), [q[0], q[1]], []),
(XGate(), [q[0]], []),
(RZXGate(-np.pi / 4), [q[0], q[1]], []),
]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
[docs] def inverse(self):
"""Return inverse ECR gate (itself)."""
return ECRGate() # self-inverse