qiskit/circuit/library/standard_gates/rx.py

# 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.

"""Rotation around the X axis."""

import math
from math import pi
from typing import Optional, Union
import numpy

from qiskit.circuit.controlledgate import ControlledGate
from qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister
from qiskit.circuit.parameterexpression import ParameterValueType


class RXGate(Gate):
    r"""Single-qubit rotation about the X axis.

    Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
    with the :meth:`~qiskit.circuit.QuantumCircuit.rx` method.

    **Circuit symbol:**

    .. parsed-literal::

             ┌───────┐
        q_0: ┤ Rx(ϴ) ├
             └───────┘

    **Matrix Representation:**

    .. math::

        \newcommand{\rotationangle}{\frac{\theta}{2}}

        RX(\theta) = \exp\left(-i \rotationangle X\right) =
            \begin{pmatrix}
                \cos\left(\rotationangle\right)   & -i\sin\left(\rotationangle\right) \\
                -i\sin\left(\rotationangle\right) & \cos\left(\rotationangle\right)
            \end{pmatrix}
    """

    def __init__(
        self, theta: ParameterValueType, label: Optional[str] = None, *, duration=None, unit="dt"
    ):
        """Create new RX gate."""
        super().__init__("rx", 1, [theta], label=label, duration=duration, unit=unit)

    def _define(self):
        """
        gate rx(theta) a {r(theta, 0) a;}
        """
        # pylint: disable=cyclic-import
        from qiskit.circuit.quantumcircuit import QuantumCircuit
        from .r import RGate

        q = QuantumRegister(1, "q")
        qc = QuantumCircuit(q, name=self.name)
        rules = [(RGate(self.params[0], 0), [q[0]], [])]
        for instr, qargs, cargs in rules:
            qc._append(instr, qargs, cargs)

        self.definition = qc

    def control(
        self,
        num_ctrl_qubits: int = 1,
        label: Optional[str] = None,
        ctrl_state: Optional[Union[str, int]] = None,
        annotated: bool = False,
    ):
        """Return a (multi-)controlled-RX gate.

        Args:
            num_ctrl_qubits: number of control qubits.
            label: An optional label for the gate [Default: ``None``]
            ctrl_state: control state expressed as integer,
                string (e.g.``'110'``), or ``None``. If ``None``, use all 1s.
            annotated: indicates whether the controlled gate can be implemented
                as an annotated gate.

        Returns:
            ControlledGate: controlled version of this gate.
        """
        if not annotated and num_ctrl_qubits == 1:
            gate = CRXGate(self.params[0], label=label, ctrl_state=ctrl_state)
            gate.base_gate.label = self.label
        else:
            gate = super().control(
                num_ctrl_qubits=num_ctrl_qubits,
                label=label,
                ctrl_state=ctrl_state,
                annotated=annotated,
            )
        return gate

    def inverse(self, annotated: bool = False):
        r"""Return inverted RX gate.

        :math:`RX(\lambda)^{\dagger} = RX(-\lambda)`

        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:`.RXGate` with an inverted parameter value.

        Returns:
            RXGate: inverse gate.
        """
        return RXGate(-self.params[0])

    def __array__(self, dtype=None):
        """Return a numpy.array for the RX gate."""
        cos = math.cos(self.params[0] / 2)
        sin = math.sin(self.params[0] / 2)
        return numpy.array([[cos, -1j * sin], [-1j * sin, cos]], dtype=dtype)

    def power(self, exponent: float, annotated: bool = False):
        (theta,) = self.params
        return RXGate(exponent * theta)

    def __eq__(self, other):
        if isinstance(other, RXGate):
            return self._compare_parameters(other)
        return False


class CRXGate(ControlledGate):
    r"""Controlled-RX gate.

    Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
    with the :meth:`~qiskit.circuit.QuantumCircuit.crx` method.

    **Circuit symbol:**

    .. parsed-literal::

        q_0: ────■────
             ┌───┴───┐
        q_1: ┤ Rx(ϴ) ├
             └───────┘

    **Matrix representation:**

    .. math::

        \newcommand{\rotationangle}{\frac{\theta}{2}}

        CRX(\theta)\ q_0, q_1 =
            I \otimes |0\rangle\langle 0| + RX(\theta) \otimes |1\rangle\langle 1| =
            \begin{pmatrix}
                1 & 0 & 0 & 0 \\
                0 & \cos\left(\rotationangle\right) & 0 & -i\sin\left(\rotationangle\right) \\
                0 & 0 & 1 & 0 \\
                0 & -i\sin\left(\rotationangle\right) & 0 & \cos\left(\rotationangle\right)
            \end{pmatrix}

    .. note::

        In Qiskit's convention, higher qubit indices are more significant
        (little endian convention). In many textbooks, controlled gates are
        presented with the assumption of more significant qubits as control,
        which in our case would be q_1. Thus a textbook matrix for this
        gate will be:

        .. parsed-literal::
                 ┌───────┐
            q_0: ┤ Rx(ϴ) ├
                 └───┬───┘
            q_1: ────■────

        .. math::

            \newcommand{\rotationangle}{\frac{\theta}{2}}

            CRX(\theta)\ q_1, q_0 =
            |0\rangle\langle0| \otimes I + |1\rangle\langle1| \otimes RX(\theta) =
                \begin{pmatrix}
                    1 & 0 & 0 & 0 \\
                    0 & 1 & 0 & 0 \\
                    0 & 0 & \cos\left(\rotationangle\right)   & -i\sin\left(\rotationangle\right) \\
                    0 & 0 & -i\sin\left(\rotationangle\right) & \cos\left(\rotationangle\right)
                \end{pmatrix}
    """

    def __init__(
        self,
        theta: ParameterValueType,
        label: Optional[str] = None,
        ctrl_state: Optional[Union[str, int]] = None,
        *,
        duration=None,
        unit="dt",
        _base_label=None,
    ):
        """Create new CRX gate."""
        super().__init__(
            "crx",
            2,
            [theta],
            num_ctrl_qubits=1,
            label=label,
            ctrl_state=ctrl_state,
            base_gate=RXGate(theta, label=_base_label),
        )

    def _define(self):
        """
        gate cu3(theta,phi,lambda) c, t
        { u1(pi/2) t;
          cx c,t;
          u3(-theta/2,0,0) t;
          cx c,t;
          u3(theta/2,-pi/2,0) t;
        }
        """
        # pylint: disable=cyclic-import
        from qiskit.circuit.quantumcircuit import QuantumCircuit
        from .u1 import U1Gate
        from .u3 import U3Gate
        from .x import CXGate

        # q_0: ─────────────■───────────────────■────────────────────
        #      ┌─────────┐┌─┴─┐┌─────────────┐┌─┴─┐┌────────────────┐
        # q_1: ┤ U1(π/2) ├┤ X ├┤ U3(0/2,0,0) ├┤ X ├┤ U3(0/2,-π/2,0) ├
        #      └─────────┘└───┘└─────────────┘└───┘└────────────────┘
        q = QuantumRegister(2, "q")
        qc = QuantumCircuit(q, name=self.name)
        rules = [
            (U1Gate(pi / 2), [q[1]], []),
            (CXGate(), [q[0], q[1]], []),
            (U3Gate(-self.params[0] / 2, 0, 0), [q[1]], []),
            (CXGate(), [q[0], q[1]], []),
            (U3Gate(self.params[0] / 2, -pi / 2, 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 CRX 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:`.CRXGate` with an inverted parameter value.

        Returns:
            CRXGate: inverse gate.
        """
        return CRXGate(-self.params[0], ctrl_state=self.ctrl_state)

    def __array__(self, dtype=None):
        """Return a numpy.array for the CRX gate."""
        half_theta = float(self.params[0]) / 2
        cos = math.cos(half_theta)
        isin = 1j * math.sin(half_theta)
        if self.ctrl_state:
            return numpy.array(
                [[1, 0, 0, 0], [0, cos, 0, -isin], [0, 0, 1, 0], [0, -isin, 0, cos]], dtype=dtype
            )
        else:
            return numpy.array(
                [[cos, 0, -isin, 0], [0, 1, 0, 0], [-isin, 0, cos, 0], [0, 0, 0, 1]], dtype=dtype
            )

    def __eq__(self, other):
        if isinstance(other, CRXGate):
            return self._compare_parameters(other) and self.ctrl_state == other.ctrl_state
        return False