Source code for qiskit.circuit.library.standard_gates.rzz

# 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 qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister


class RZZGate(Gate):
    r"""A parameteric 2-qubit :math:`Z \otimes Z` interaction (rotation about ZZ).

    This gate is symmetric, and is maximally entangling at :math:`\theta = \pi/2`.

    **Circuit Symbol:**

    .. parsed-literal::

        q_0: ───■────
                │zz(θ)
        q_1: ───■────

    **Matrix Representation:**

    .. math::

        \newcommand{\th}{\frac{\theta}{2}}

        R_{ZZ}(\theta) = exp(-i \th Z{\otimes}Z) =
            \begin{pmatrix}
                e^{-i \th} & 0 & 0 & 0 \\
                0 & e^{i \th} & 0 & 0 \\
                0 & 0 & e^{i \th} & 0 \\
                0 & 0 & 0 & e^{-i \th}
            \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}(\theta = \frac{\pi}{2}) = \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):
        """Create new RZZ gate."""
        super().__init__('rzz', 2, [theta])

    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 = 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):
        """Return inverse RZZ gate (i.e. with the negative rotation angle)."""
        return RZZGate(-self.params[0])

    def to_matrix(self):
        """Return a numpy.array for the RZZ gate."""
        import numpy
        itheta2 = 1j * float(self.params[0]) / 2
        return numpy.array([[numpy.exp(-itheta2), 0, 0, 0],
                            [0, numpy.exp(itheta2), 0, 0],
                            [0, 0, numpy.exp(itheta2), 0],
                            [0, 0, 0, numpy.exp(-itheta2)]], dtype=complex)