Source code for pennylane.ops.op_math.controlled_ops

# Copyright 2018-2024 Xanadu Quantum Technologies Inc.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

#     http://www.apache.org/licenses/LICENSE-2.0

# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
This submodule contains controlled operators based on the ControlledOp class.
"""
# pylint: disable=no-value-for-parameter, arguments-differ, arguments-renamed
import warnings
from collections.abc import Iterable
from functools import lru_cache

import numpy as np
from scipy.linalg import block_diag

import pennylane as qml
from pennylane.operation import AnyWires, Wires
from pennylane.ops.qubit.parametric_ops_single_qubit import stack_last
from pennylane.wires import WiresLike

from .controlled import ControlledOp
from .controlled_decompositions import decompose_mcx

INV_SQRT2 = 1 / qml.math.sqrt(2)


# pylint: disable=too-few-public-methods
class ControlledQubitUnitary(ControlledOp):
    r"""ControlledQubitUnitary(U, control_wires, wires, control_values)
    Apply an arbitrary fixed unitary to ``wires`` with control from the ``control_wires``.

    In addition to default ``Operation`` instance attributes, the following are
    available for ``ControlledQubitUnitary``:

    * ``control_wires``: wires that act as control for the operation
    * ``control_values``: the state on which to apply the controlled operation (see below)
    * ``target_wires``: the wires the unitary matrix will be applied to
    * ``work_wires``: wires made use of during the decomposition of the operation into native operations

    **Details:**

    * Number of wires: Any (the operation can act on any number of wires)
    * Number of parameters: 1
    * Number of dimensions per parameter: (2,)
    * Gradient recipe: None

    Args:
        base (Union[array[complex], QubitUnitary]): square unitary matrix or a QubitUnitary
            operation. If passing a matrix, this will be used to construct a QubitUnitary
            operator that will be used as the base operator. If providing a ``qml.QubitUnitary``,
            this will be used as the base directly.
        control_wires (Union[Wires, Sequence[int], or int]): the control wire(s)
        wires (Union[Wires, Sequence[int], or int]): the wire(s) the unitary acts on
            (optional if U is provided as a QubitUnitary)
        control_values (List[int, bool]): a list providing the state of the control qubits to
            control on (default is the all 1s state)
        unitary_check (bool): whether to check whether an array U is unitary when creating the
            operator (default False)
        work_wires (Union[Wires, Sequence[int], or int]): ancillary wire(s) that may be utilized in during
            the decomposition of the operator into native operations.

    **Example**

    The following shows how a single-qubit unitary can be applied to wire ``2`` with control on
    both wires ``0`` and ``1``:

    >>> U = np.array([[ 0.94877869,  0.31594146], [-0.31594146,  0.94877869]])
    >>> qml.ControlledQubitUnitary(U, control_wires=[0, 1], wires=2)
    Controlled(QubitUnitary(array([[ 0.94877869,  0.31594146],
       [-0.31594146,  0.94877869]]), wires=[2]), control_wires=[0, 1])

    Alternatively, the same operator can be constructed with a QubitUnitary:

    >>> base = qml.QubitUnitary(U, wires=2)
    >>> qml.ControlledQubitUnitary(base, control_wires=[0, 1])
    Controlled(QubitUnitary(array([[ 0.94877869,  0.31594146],
       [-0.31594146,  0.94877869]]), wires=[2]), control_wires=[0, 1])

    Typically, controlled operations apply a desired gate if the control qubits
    are all in the state :math:`\vert 1\rangle`. However, there are some situations where
    it is necessary to apply a gate conditioned on all qubits being in the
    :math:`\vert 0\rangle` state, or a mix of the two.

    The state on which to control can be changed by passing a string of bits to
    `control_values`. For example, if we want to apply a single-qubit unitary to
    wire ``3`` conditioned on three wires where the first is in state ``0``, the
    second is in state ``1``, and the third in state ``1``, we can write:

    >>> qml.ControlledQubitUnitary(U, control_wires=[0, 1, 2], wires=3, control_values=[0, 1, 1])

    or

    >>> qml.ControlledQubitUnitary(U, control_wires=[0, 1, 2], wires=3, control_values=[False, True, True])
    """

    num_wires = AnyWires
    """int: Number of wires that the operator acts on."""

    num_params = 1
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = (2,)
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    grad_method = None
    """Gradient computation method."""

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(
            data[0], control_wires=metadata[0], control_values=metadata[1], work_wires=metadata[2]
        )

    # pylint: disable=arguments-differ, too-many-arguments, unused-argument, too-many-positional-arguments
    @classmethod
    def _primitive_bind_call(
        cls,
        base,
        control_wires: WiresLike,
        wires: WiresLike = (),
        control_values=None,
        unitary_check=False,
        work_wires: WiresLike = (),
    ):
        wires = Wires(() if wires is None else wires)
        work_wires = Wires(() if work_wires is None else work_wires)

        if hasattr(base, "wires") and len(wires) != 0:
            warnings.warn(
                "base operator already has wires; values specified through wires kwarg will be ignored."
            )
            wires = Wires(())

        all_wires = control_wires + wires
        return cls._primitive.bind(
            base, control_wires=all_wires, control_values=control_values, work_wires=work_wires
        )

    # pylint: disable=too-many-arguments,too-many-positional-arguments
    def __init__(
        self,
        base,
        control_wires: WiresLike,
        wires: WiresLike = (),
        control_values=None,
        unitary_check=False,
        work_wires: WiresLike = (),
    ):
        wires = Wires(() if wires is None else wires)
        work_wires = Wires(() if work_wires is None else work_wires)
        control_wires = Wires(control_wires)

        if hasattr(base, "wires") and len(wires) != 0:
            warnings.warn(
                "base operator already has wires; values specified through wires kwarg will be ignored."
            )
            wires = Wires(())

        if isinstance(base, Iterable):
            if len(wires) == 0:
                if len(control_wires) > 1:
                    num_base_wires = int(qml.math.log2(qml.math.shape(base)[-1]))
                    wires = control_wires[-num_base_wires:]
                    control_wires = control_wires[:-num_base_wires]
                else:
                    raise TypeError(
                        "Must specify a set of wires. None is not a valid `wires` label."
                    )
            # We use type.__call__ instead of calling the class directly so that we don't bind the
            # operator primitive when new program capture is enabled
            base = type.__call__(qml.QubitUnitary, base, wires=wires, unitary_check=unitary_check)

        super().__init__(
            base,
            control_wires,
            control_values=control_values,
            work_wires=work_wires,
        )
        self._name = "ControlledQubitUnitary"

    def _controlled(self, wire):
        ctrl_wires = wire + self.control_wires
        values = None if self.control_values is None else [True] + self.control_values
        return ControlledQubitUnitary(
            self.base,
            control_wires=ctrl_wires,
            control_values=values,
            work_wires=self.work_wires,
        )

    @property
    def has_decomposition(self):
        if not super().has_decomposition:
            return False
        with qml.QueuingManager.stop_recording():
            # we know this is using try-except as logical control, but are favouring
            # certainty in it being correct over explicitness in an edge case.
            try:
                self.decomposition()
            except qml.operation.DecompositionUndefinedError:
                return False
        return True


class CH(ControlledOp):
    r"""CH(wires)
    The controlled-Hadamard operator

    .. math:: CH = \begin{bmatrix}
            1 & 0 & 0 & 0 \\
            0 & 1 & 0 & 0 \\
            0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
            0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
        \end{bmatrix}.

    .. note:: The first wire provided corresponds to the **control qubit**.

    **Details:**

    * Number of wires: 2
    * Number of parameters: 0

    Args:
        wires (Sequence[int]): the wires the operation acts on
    """

    num_wires = 2
    """int: Number of wires that the operation acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CH"

    def _flatten(self):
        return tuple(), (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(metadata[0])

    @classmethod
    def _primitive_bind_call(cls, wires, id=None):
        return cls._primitive.bind(*wires, n_wires=2)

    def __init__(self, wires, id=None):
        control_wires = wires[:1]
        target_wires = wires[1:]

        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.Hadamard, wires=target_wires)
        super().__init__(base, control_wires, id=id)

    def __repr__(self):
        return f"CH(wires={self.wires.tolist()})"

    @staticmethod
    @lru_cache()
    def compute_matrix():  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CH.matrix`


        Returns:
            ndarray: matrix

        **Example**

        >>> print(qml.CH.compute_matrix())
        [[ 1.          0.          0.          0.        ]
         [ 0.          1.          0.          0.        ]
         [ 0.          0.          0.70710678  0.70710678]
         [ 0.          0.          0.70710678 -0.70710678]]
        """
        return np.array(
            [
                [1, 0, 0, 0],
                [0, 1, 0, 0],
                [0, 0, INV_SQRT2, INV_SQRT2],
                [0, 0, INV_SQRT2, -INV_SQRT2],
            ]
        )

    @staticmethod
    def compute_decomposition(wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method).


        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.CH.decomposition`.

        Args:
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> print(qml.CH.compute_decomposition([0, 1]))
        [RY(-0.7853981633974483, wires=[1]), CZ(wires=[0, 1]), RY(0.7853981633974483, wires=[1])]

        """
        return [
            qml.RY(-np.pi / 4, wires=wires[1]),
            qml.CZ(wires=wires),
            qml.RY(+np.pi / 4, wires=wires[1]),
        ]


class CY(ControlledOp):
    r"""CY(wires)
    The controlled-Y operator

    .. math:: CY = \begin{bmatrix}
            1 & 0 & 0 & 0 \\
            0 & 1 & 0 & 0\\
            0 & 0 & 0 & -i\\
            0 & 0 & i & 0
        \end{bmatrix}.

    .. note:: The first wire provided corresponds to the **control qubit**.

    **Details:**

    * Number of wires: 2
    * Number of parameters: 0

    Args:
        wires (Sequence[int]): the wires the operation acts on
        id (str): custom label given to an operator instance,
            can be useful for some applications where the instance has to be identified.
    """

    num_wires = 2
    """int: Number of wires that the operator acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CY"

    def _flatten(self):
        return tuple(), (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(metadata[0])

    @classmethod
    def _primitive_bind_call(cls, wires, id=None):
        return cls._primitive.bind(*wires, n_wires=2)

    def __init__(self, wires, id=None):
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.Y, wires=wires[1:])
        super().__init__(base, wires[:1], id=id)

    def __repr__(self):
        return f"CY(wires={self.wires.tolist()})"

    @staticmethod
    @lru_cache()
    def compute_matrix():  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CY.matrix`


        Returns:
            ndarray: matrix

        **Example**

        >>> print(qml.CY.compute_matrix())
        [[ 1.+0.j  0.+0.j  0.+0.j  0.+0.j]
         [ 0.+0.j  1.+0.j  0.+0.j  0.+0.j]
         [ 0.+0.j  0.+0.j  0.+0.j -0.-1.j]
         [ 0.+0.j  0.+0.j  0.+1.j  0.+0.j]]
        """
        return np.array(
            [
                [1, 0, 0, 0],
                [0, 1, 0, 0],
                [0, 0, 0, -1j],
                [0, 0, 1j, 0],
            ]
        )

    @staticmethod
    def compute_decomposition(wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method).


        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.CY.decomposition`.

        Args:
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> print(qml.CY.compute_decomposition([0, 1]))
        [CRY(3.141592653589793, wires=[0, 1])), S(0)]

        """
        return [qml.CRY(np.pi, wires=wires), qml.S(wires=wires[0])]


class CZ(ControlledOp):
    r"""CZ(wires)
    The controlled-Z operator

    .. math:: CZ = \begin{bmatrix}
            1 & 0 & 0 & 0 \\
            0 & 1 & 0 & 0\\
            0 & 0 & 1 & 0\\
            0 & 0 & 0 & -1
        \end{bmatrix}.

    .. note:: The first wire provided corresponds to the **control qubit**.

    **Details:**

    * Number of wires: 2
    * Number of parameters: 0

    Args:
        wires (Sequence[int]): the wires the operation acts on
    """

    num_wires = 2
    """int: Number of wires that the operator acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CZ"

    def _flatten(self):
        return tuple(), (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(metadata[0])

    @classmethod
    def _primitive_bind_call(cls, wires, id=None):
        return cls._primitive.bind(*wires, n_wires=2)

    def __init__(self, wires, id=None):
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.Z, wires=wires[1:])
        super().__init__(base, wires[:1], id=id)

    def __repr__(self):
        return f"CZ(wires={self.wires.tolist()})"

    @staticmethod
    @lru_cache()
    def compute_matrix():  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CZ.matrix`

        Returns:
            ndarray: matrix

        **Example**

        >>> print(qml.CZ.compute_matrix())
        [[ 1  0  0  0]
         [ 0  1  0  0]
         [ 0  0  1  0]
         [ 0  0  0 -1]]
        """
        return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1]])

    def _controlled(self, wire):
        return qml.CCZ(wires=wire + self.wires)

    @staticmethod
    def compute_decomposition(wires):  # pylint: disable=arguments-differ
        return [qml.ControlledPhaseShift(np.pi, wires=wires)]


class CSWAP(ControlledOp):
    r"""CSWAP(wires)
    The controlled-swap operator

    .. math:: CSWAP = \begin{bmatrix}
            1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
            0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
            0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
            0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
            0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
            0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
            0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
            0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
        \end{bmatrix}.

    .. note:: The first wire provided corresponds to the **control qubit**.

    **Details:**

    * Number of wires: 3
    * Number of parameters: 0

    Args:
        wires (Sequence[int]): the wires the operation acts on
    """

    num_wires = 3
    """int : Number of wires that the operation acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CSWAP"

    def _flatten(self):
        return tuple(), (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(metadata[0])

    @classmethod
    def _primitive_bind_call(cls, wires, id=None):
        return cls._primitive.bind(*wires, n_wires=3)

    def __init__(self, wires, id=None):
        control_wires = wires[:1]
        target_wires = wires[1:]

        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.SWAP, wires=target_wires)
        super().__init__(base, control_wires, id=id)

    def __repr__(self):
        return f"CSWAP(wires={self.wires.tolist()})"

    @staticmethod
    @lru_cache()
    def compute_matrix():  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CSWAP.matrix`

        Returns:
            ndarray: matrix

        **Example**

        >>> print(qml.CSWAP.compute_matrix())
        [[1 0 0 0 0 0 0 0]
         [0 1 0 0 0 0 0 0]
         [0 0 1 0 0 0 0 0]
         [0 0 0 1 0 0 0 0]
         [0 0 0 0 1 0 0 0]
         [0 0 0 0 0 0 1 0]
         [0 0 0 0 0 1 0 0]
         [0 0 0 0 0 0 0 1]]
        """
        return np.array(
            [
                [1, 0, 0, 0, 0, 0, 0, 0],
                [0, 1, 0, 0, 0, 0, 0, 0],
                [0, 0, 1, 0, 0, 0, 0, 0],
                [0, 0, 0, 1, 0, 0, 0, 0],
                [0, 0, 0, 0, 1, 0, 0, 0],
                [0, 0, 0, 0, 0, 0, 1, 0],
                [0, 0, 0, 0, 0, 1, 0, 0],
                [0, 0, 0, 0, 0, 0, 0, 1],
            ]
        )

    @staticmethod
    def compute_decomposition(wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method).

        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.CSWAP.decomposition`.

        Args:
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> print(qml.CSWAP.compute_decomposition((0,1,2)))
        [Toffoli(wires=[0, 2, 1]), Toffoli(wires=[0, 1, 2]), Toffoli(wires=[0, 2, 1])]

        """
        decomp_ops = [
            qml.Toffoli(wires=[wires[0], wires[2], wires[1]]),
            qml.Toffoli(wires=[wires[0], wires[1], wires[2]]),
            qml.Toffoli(wires=[wires[0], wires[2], wires[1]]),
        ]
        return decomp_ops


class CCZ(ControlledOp):
    r"""CCZ(wires)
    CCZ (controlled-controlled-Z) gate.

    .. math::

        CCZ =
        \begin{pmatrix}
        1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
        0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
        0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
        0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
        0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
        0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
        0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
        \end{pmatrix}

    **Details:**

    * Number of wires: 3
    * Number of parameters: 0

    Args:
        wires (Sequence[int]): the subsystem the gate acts on
    """

    @classmethod
    def _primitive_bind_call(cls, wires, id=None):
        return cls._primitive.bind(*wires, n_wires=3)

    def _flatten(self):
        return tuple(), (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(metadata[0])

    num_wires = 3
    """int: Number of wires that the operator acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CCZ"

    def __init__(self, wires, id=None):
        control_wires = wires[:2]
        target_wires = wires[2:]

        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.Z, wires=target_wires)
        super().__init__(base, control_wires, id=id)

    def __repr__(self):
        return f"CCZ(wires={self.wires.tolist()})"

    @staticmethod
    @lru_cache()
    def compute_matrix():  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CCZ.matrix`


        Returns:
            ndarray: matrix

        **Example**

        >>> print(qml.CCZ.compute_matrix())
        [[1 0 0 0 0 0 0 0]
         [0 1 0 0 0 0 0 0]
         [0 0 1 0 0 0 0 0]
         [0 0 0 1 0 0 0 0]
         [0 0 0 0 1 0 0 0]
         [0 0 0 0 0 1 0 0]
         [0 0 0 0 0 0 1 0]
         [0 0 0 0 0 0 0 -1]]
        """
        return np.array(
            [
                [1, 0, 0, 0, 0, 0, 0, 0],
                [0, 1, 0, 0, 0, 0, 0, 0],
                [0, 0, 1, 0, 0, 0, 0, 0],
                [0, 0, 0, 1, 0, 0, 0, 0],
                [0, 0, 0, 0, 1, 0, 0, 0],
                [0, 0, 0, 0, 0, 1, 0, 0],
                [0, 0, 0, 0, 0, 0, 1, 0],
                [0, 0, 0, 0, 0, 0, 0, -1],
            ]
        )

    @staticmethod
    def compute_decomposition(wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method).

        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.Toffoli.decomposition`.

        Args:
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> qml.CCZ.compute_decomposition((0,1,2))
        [CNOT(wires=[1, 2]),
         Adjoint(T(2)),
         CNOT(wires=[0, 2]),
         T(2),
         CNOT(wires=[1, 2]),
         Adjoint(T(2)),
         CNOT(wires=[0, 2]),
         T(2),
         T(1),
         CNOT(wires=[0, 1]),
         H(2),
         T(0),
         Adjoint(T(1)),
         CNOT(wires=[0, 1]),
         H(2)]

        """
        return [
            qml.CNOT(wires=[wires[1], wires[2]]),
            qml.adjoint(qml.T(wires=wires[2])),
            qml.CNOT(wires=[wires[0], wires[2]]),
            qml.T(wires=wires[2]),
            qml.CNOT(wires=[wires[1], wires[2]]),
            qml.adjoint(qml.T(wires=wires[2])),
            qml.CNOT(wires=[wires[0], wires[2]]),
            qml.T(wires=wires[2]),
            qml.T(wires=wires[1]),
            qml.CNOT(wires=[wires[0], wires[1]]),
            qml.Hadamard(wires=wires[2]),
            qml.T(wires=wires[0]),
            qml.adjoint(qml.T(wires=wires[1])),
            qml.CNOT(wires=[wires[0], wires[1]]),
            qml.Hadamard(wires=wires[2]),
        ]


class CNOT(ControlledOp):
    r"""CNOT(wires)
    The controlled-NOT operator

    .. math:: CNOT = \begin{bmatrix}
        1 & 0 & 0 & 0 \\
        0 & 1 & 0 & 0\\
        0 & 0 & 0 & 1\\
        0 & 0 & 1 & 0
        \end{bmatrix}.

    .. note:: The first wire provided corresponds to the **control qubit**.

    **Details:**

    * Number of wires: 2
    * Number of parameters: 0

    Args:
        wires (Sequence[int]): the wires the operation acts on
    """

    num_wires = 2
    """int: Number of wires that the operator acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CNOT"

    def _flatten(self):
        return tuple(), (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(metadata[0])

    @classmethod
    def _primitive_bind_call(cls, wires, id=None):
        return cls._primitive.bind(*wires, n_wires=2)

    def __init__(self, wires, id=None):
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.X, wires=wires[1:])
        super().__init__(base, wires[:1], id=id)

    @property
    def has_decomposition(self):
        return False

    @staticmethod
    def compute_decomposition(*params, wires=None, **hyperparameters):  # -> List["Operator"]:
        r"""Representation of the operator as a product of other operators (static method).

        .. math:: O = O_1 O_2 \dots O_n.

        .. note::
            Operations making up the decomposition should be queued within the
            ``compute_decomposition`` method.

        .. seealso:: :meth:`~.Operator.decomposition`.

        Args:
            *params (list): trainable parameters of the operator, as stored in the ``parameters`` attribute
            wires (Iterable[Any], Wires): wires that the operator acts on
            **hyperparams (dict): non-trainable hyperparameters of the operator, as stored in the ``hyperparameters`` attribute

        Raises:
            qml.DecompositionUndefinedError
        """
        raise qml.operation.DecompositionUndefinedError

    def __repr__(self):
        return f"CNOT(wires={self.wires.tolist()})"

    @staticmethod
    @lru_cache()
    def compute_matrix():  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CNOT.matrix`


        Returns:
            ndarray: matrix

        **Example**

        >>> print(qml.CNOT.compute_matrix())
        [[1 0 0 0]
         [0 1 0 0]
         [0 0 0 1]
         [0 0 1 0]]
        """
        return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]])

    def _controlled(self, wire):
        return qml.Toffoli(wires=wire + self.wires)


class Toffoli(ControlledOp):
    r"""Toffoli(wires)
    Toffoli (controlled-controlled-X) gate.

    .. math::

        Toffoli =
        \begin{pmatrix}
        1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
        0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
        0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
        0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
        0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
        0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
        0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
        0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
        \end{pmatrix}

    **Details:**

    * Number of wires: 3
    * Number of parameters: 0

    Args:
        wires (Sequence[int]): the subsystem the gate acts on
    """

    num_wires = 3
    """int: Number of wires that the operator acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "Toffoli"

    def _flatten(self):
        return tuple(), (self.wires,)

    @classmethod
    def _unflatten(cls, _, metadata):
        return cls(metadata[0])

    @classmethod
    def _primitive_bind_call(cls, wires, id=None):
        return cls._primitive.bind(*wires, n_wires=3)

    def __init__(self, wires, id=None):
        control_wires = wires[:2]
        target_wires = wires[2:]
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.X, wires=target_wires)
        super().__init__(base, control_wires, id=id)

    def __repr__(self):
        return f"Toffoli(wires={self.wires.tolist()})"

    @staticmethod
    @lru_cache()
    def compute_matrix():  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.Toffoli.matrix`


        Returns:
            ndarray: matrix

        **Example**

        >>> print(qml.Toffoli.compute_matrix())
        [[1 0 0 0 0 0 0 0]
         [0 1 0 0 0 0 0 0]
         [0 0 1 0 0 0 0 0]
         [0 0 0 1 0 0 0 0]
         [0 0 0 0 1 0 0 0]
         [0 0 0 0 0 1 0 0]
         [0 0 0 0 0 0 0 1]
         [0 0 0 0 0 0 1 0]]
        """
        return np.array(
            [
                [1, 0, 0, 0, 0, 0, 0, 0],
                [0, 1, 0, 0, 0, 0, 0, 0],
                [0, 0, 1, 0, 0, 0, 0, 0],
                [0, 0, 0, 1, 0, 0, 0, 0],
                [0, 0, 0, 0, 1, 0, 0, 0],
                [0, 0, 0, 0, 0, 1, 0, 0],
                [0, 0, 0, 0, 0, 0, 0, 1],
                [0, 0, 0, 0, 0, 0, 1, 0],
            ]
        )

    @staticmethod
    def compute_decomposition(wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method).

        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.Toffoli.decomposition`.

        Args:
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> qml.Toffoli.compute_decomposition((0,1,2))
        [H(2),
         CNOT(wires=[1, 2]),
         Adjoint(T(2)),
         CNOT(wires=[0, 2]),
         T(2),
         CNOT(wires=[1, 2]),
         Adjoint(T(2)),
         CNOT(wires=[0, 2]),
         T(2),
         T(1),
         CNOT(wires=[0, 1]),
         H(2),
         T(0),
         Adjoint(T(1)),
         CNOT(wires=[0, 1])]

        """
        return [
            qml.Hadamard(wires=wires[2]),
            CNOT(wires=[wires[1], wires[2]]),
            qml.adjoint(qml.T(wires=wires[2])),
            CNOT(wires=[wires[0], wires[2]]),
            qml.T(wires=wires[2]),
            CNOT(wires=[wires[1], wires[2]]),
            qml.adjoint(qml.T(wires=wires[2])),
            CNOT(wires=[wires[0], wires[2]]),
            qml.T(wires=wires[2]),
            qml.T(wires=wires[1]),
            CNOT(wires=[wires[0], wires[1]]),
            qml.Hadamard(wires=wires[2]),
            qml.T(wires=wires[0]),
            qml.adjoint(qml.T(wires=wires[1])),
            CNOT(wires=[wires[0], wires[1]]),
        ]


def _check_and_convert_control_values(control_values, control_wires):
    if isinstance(control_values, str):
        # Make sure all values are either 0 or 1
        if not set(control_values).issubset({"1", "0"}):
            raise ValueError("String of control values can contain only '0' or '1'.")

        control_values = [int(x) for x in control_values]

    if control_values is None:
        return [1] * len(control_wires)

    if len(control_values) != len(control_wires):
        raise ValueError("Length of control values must equal number of control wires.")

    return control_values


class MultiControlledX(ControlledOp):
    r"""MultiControlledX(control_wires, wires, control_values, work_wires)
    Apply a Pauli X gate controlled on an arbitrary computational basis state.

    **Details:**

    * Number of wires: Any (the operation can act on any number of wires)
    * Number of parameters: 0
    * Gradient recipe: None

    Args:
        control_wires (Union[Wires, Sequence[int], or int]): Deprecated way to indicate the control wires.
            Now users should use "wires" to indicate both the control wires and the target wire.
        wires (Union[Wires, Sequence[int], or int]): control wire(s) followed by a single target wire where
            the operation acts on
        control_values (Union[bool, list[bool], int, list[int]]): The value(s) the control wire(s)
                should take. Integers other than 0 or 1 will be treated as ``int(bool(x))``.
        work_wires (Union[Wires, Sequence[int], or int]): optional work wires used to decompose
            the operation into a series of Toffoli gates


    .. note::

        If ``MultiControlledX`` is not supported on the targeted device, PennyLane will decompose
        the operation into :class:`~.Toffoli` and/or :class:`~.CNOT` gates. When controlling on
        three or more wires, the Toffoli-based decompositions described in Lemmas 7.2 and 7.3 of
        `Barenco et al. <https://arxiv.org/abs/quant-ph/9503016>`__ will be used. These methods
        require at least one work wire.

        The number of work wires provided determines the decomposition method used and the resulting
        number of Toffoli gates required. When ``MultiControlledX`` is controlling on :math:`n`
        wires:

        #. If at least :math:`n - 2` work wires are provided, the decomposition in Lemma 7.2 will be
           applied using the first :math:`n - 2` work wires.
        #. If fewer than :math:`n - 2` work wires are provided, a combination of Lemmas 7.3 and 7.2
           will be applied using only the first work wire.

        These methods present a tradeoff between qubit number and depth. The method in point 1
        requires fewer Toffoli gates but a greater number of qubits.

        Note that the state of the work wires before and after the decomposition takes place is
        unchanged.

    """

    is_self_inverse = True
    """bool: Whether or not the operator is self-inverse."""

    num_wires = AnyWires
    """int: Number of wires the operation acts on."""

    num_params = 0
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = ()
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "MultiControlledX"

    def _flatten(self):
        return (), (self.wires, tuple(self.control_values), self.work_wires)

    @classmethod
    def _unflatten(cls, _, metadata):
        return cls(wires=metadata[0], control_values=metadata[1], work_wires=metadata[2])

    # pylint: disable=arguments-differ
    @classmethod
    def _primitive_bind_call(cls, wires, control_values=None, work_wires=None, id=None):
        return cls._primitive.bind(
            *wires, n_wires=len(wires), control_values=control_values, work_wires=work_wires
        )

    # pylint: disable=too-many-arguments
    def __init__(
        self,
        control_wires: WiresLike = (),
        wires: WiresLike = (),
        control_values=None,
        work_wires: WiresLike = (),
    ):
        control_wires = Wires(() if control_wires is None else control_wires)
        wires = Wires(() if wires is None else wires)
        work_wires = Wires(() if work_wires is None else work_wires)

        # First raise deprecation warnings regardless of the validity of other arguments
        if isinstance(control_values, str):
            warnings.warn(
                "Specifying control values using a bitstring is deprecated, and will not be "
                "supported in future releases, Use a list of booleans or integers instead.",
                qml.PennyLaneDeprecationWarning,
            )
        if len(control_wires) > 0:
            warnings.warn(
                "The control_wires keyword for MultiControlledX is deprecated, and will "
                "be removed soon. Use wires = (*control_wires, target_wire) instead.",
                UserWarning,
            )

        if len(wires) == 0:
            raise ValueError("Must specify the wires where the operation acts on")

        if len(control_wires) > 0:
            if len(wires) != 1:
                raise ValueError("MultiControlledX accepts a single target wire.")
        else:
            if len(wires) < 2:
                raise ValueError(
                    f"MultiControlledX: wrong number of wires. {len(wires)} wire(s) given. "
                    f"Need at least 2."
                )
            control_wires = wires[:-1]
            wires = wires[-1:]

        control_values = _check_and_convert_control_values(control_values, control_wires)

        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.X, wires=wires)
        super().__init__(
            base,
            control_wires=control_wires,
            control_values=control_values,
            work_wires=work_wires,
        )

    def __repr__(self):
        return (
            f"MultiControlledX(wires={self.wires.tolist()}, control_values={self.control_values})"
        )

    @property
    def wires(self):
        return self.control_wires + self.target_wires

    # pylint: disable=unused-argument, arguments-differ
    @staticmethod
    def compute_matrix(control_wires: WiresLike, control_values=None, **kwargs):
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.MultiControlledX.matrix`

        Args:
            control_wires (Any or Iterable[Any]): wires to place controls on
            control_values (Union[bool, list[bool], int, list[int]]): The value(s) the control wire(s)
                should take. Integers other than 0 or 1 will be treated as ``int(bool(x))``.

        Returns:
            tensor_like: matrix representation

        **Example**

        >>> print(qml.MultiControlledX.compute_matrix([0], [1]))
        [[1. 0. 0. 0.]
         [0. 1. 0. 0.]
         [0. 0. 0. 1.]
         [0. 0. 1. 0.]]
        >>> print(qml.MultiControlledX.compute_matrix([1], [0]))
        [[0. 1. 0. 0.]
         [1. 0. 0. 0.]
         [0. 0. 1. 0.]
         [0. 0. 0. 1.]]

        """

        control_values = _check_and_convert_control_values(control_values, control_wires)
        padding_left = sum(2**i * int(val) for i, val in enumerate(reversed(control_values))) * 2
        padding_right = 2 ** (len(control_wires) + 1) - 2 - padding_left
        return block_diag(np.eye(padding_left), qml.X.compute_matrix(), np.eye(padding_right))

    def matrix(self, wire_order=None):
        canonical_matrix = self.compute_matrix(self.control_wires, self.control_values)
        wire_order = wire_order or self.wires
        return qml.math.expand_matrix(canonical_matrix, wires=self.wires, wire_order=wire_order)

    # pylint: disable=unused-argument, arguments-differ
    @staticmethod
    def compute_decomposition(
        wires: WiresLike = None, work_wires: WiresLike = None, control_values=None, **kwargs
    ):
        r"""Representation of the operator as a product of other operators (static method).

        .. math:: O = O_1 O_2 \dots O_n.

        .. seealso:: :meth:`~.MultiControlledX.decomposition`.

        Args:
            wires (Iterable[Any] or Wires): wires that the operation acts on
            work_wires (Wires): optional work wires used to decompose
                the operation into a series of Toffoli gates.
            control_values (Union[bool, list[bool], int, list[int]]): The value(s) the control wire(s)
                should take. Integers other than 0 or 1 will be treated as ``int(bool(x))``.

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> print(qml.MultiControlledX.compute_decomposition(
        ...     wires=[0,1,2,3], control_values=[1,1,1], work_wires=qml.wires.Wires("aux")))
        [Toffoli(wires=[2, 'aux', 3]),
        Toffoli(wires=[0, 1, 'aux']),
        Toffoli(wires=[2, 'aux', 3]),
        Toffoli(wires=[0, 1, 'aux'])]

        """
        wires = Wires(() if wires is None else wires)

        if len(wires) < 2:
            raise ValueError(f"Wrong number of wires. {len(wires)} given. Need at least 2.")

        target_wire = wires[-1]
        control_wires = wires[:-1]

        if control_values is None:
            control_values = [True] * len(control_wires)

        work_wires = work_wires or []

        flips1 = [qml.X(w) for w, val in zip(control_wires, control_values) if not val]

        decomp = decompose_mcx(control_wires, target_wire, work_wires)

        flips2 = [qml.X(w) for w, val in zip(control_wires, control_values) if not val]

        return flips1 + decomp + flips2

    def decomposition(self):
        return self.compute_decomposition(self.wires, self.work_wires, self.control_values)


class CRX(ControlledOp):
    r"""The controlled-RX operator

    .. math::

        \begin{align}
            CR_x(\phi) &=
            \begin{bmatrix}
            & 1 & 0 & 0 & 0 \\
            & 0 & 1 & 0 & 0\\
            & 0 & 0 & \cos(\phi/2) & -i\sin(\phi/2)\\
            & 0 & 0 & -i\sin(\phi/2) & \cos(\phi/2)
            \end{bmatrix}.
        \end{align}

    **Details:**

    * Number of wires: 2
    * Number of parameters: 1
    * Number of dimensions per parameter: (0,)
    * Gradient recipe: The controlled-RX operator satisfies a four-term parameter-shift rule
      (see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):

      .. math::

          \frac{d}{d\phi}f(CR_x(\phi)) = c_+ \left[f(CR_x(\phi+a)) - f(CR_x(\phi-a))\right] - c_- \left[f(CR_x(\phi+b)) - f(CR_x(\phi-b))\right]

      where :math:`f` is an expectation value depending on :math:`CR_x(\phi)`, and

      - :math:`a = \pi/2`
      - :math:`b = 3\pi/2`
      - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`

    Args:
        phi (float): rotation angle :math:`\phi`
        wires (Sequence[int]): the wire the operation acts on
        id (str or None): String representing the operation (optional)
    """

    num_wires = 2
    """int: Number of wires that the operation acts on."""

    num_params = 1
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = (0,)
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CRX"
    parameter_frequencies = [(0.5, 1.0)]

    def __init__(self, phi, wires: WiresLike, id=None):
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.RX, phi, wires=wires[1:])
        super().__init__(base, control_wires=wires[:1], id=id)

    def __repr__(self):
        return f"CRX({self.data[0]}, wires={self.wires.tolist()})"

    def _flatten(self):
        return self.data, (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(*data, wires=metadata[0])

    @classmethod
    def _primitive_bind_call(cls, phi, wires: WiresLike, id=None):
        return cls._primitive.bind(phi, *wires, n_wires=len(wires))

    @staticmethod
    def compute_matrix(theta):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CRX.matrix`

        Args:
            theta (tensor_like or float): rotation angle

        Returns:
            tensor_like: canonical matrix

        **Example**

        >>> qml.CRX.compute_matrix(torch.tensor(0.5))
        tensor([[1.0+0.0j, 0.0+0.0j,    0.0+0.0j,    0.0+0.0j],
                [0.0+0.0j, 1.0+0.0j,    0.0+0.0j,    0.0+0.0j],
                [0.0+0.0j, 0.0+0.0j, 0.9689+0.0j, 0.0-0.2474j],
                [0.0+0.0j, 0.0+0.0j, 0.0-0.2474j, 0.9689+0.0j]])
        """

        interface = qml.math.get_interface(theta)

        c = qml.math.cos(theta / 2)
        s = qml.math.sin(theta / 2)

        if interface == "tensorflow":
            c = qml.math.cast_like(c, 1j)
            s = qml.math.cast_like(s, 1j)

        # The following avoids casting an imaginary quantity to reals when back propagating
        c = (1 + 0j) * c
        js = -1j * s
        ones = qml.math.ones_like(js)
        zeros = qml.math.zeros_like(js)
        matrix = [
            [ones, zeros, zeros, zeros],
            [zeros, ones, zeros, zeros],
            [zeros, zeros, c, js],
            [zeros, zeros, js, c],
        ]

        return qml.math.stack([stack_last(row) for row in matrix], axis=-2)

    @staticmethod
    def compute_decomposition(phi, wires: WiresLike):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method). :

        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.CRot.decomposition`.

        Args:
            phi (float): rotation angle :math:`\phi`
            wires (Iterable, Wires): the wires the operation acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> qml.CRX.compute_decomposition(1.2, wires=(0,1))
        [RZ(1.5707963267948966, wires=[1]),
        RY(0.6, wires=[1]),
        CNOT(wires=[0, 1]),
        RY(-0.6, wires=[1]),
        CNOT(wires=[0, 1]),
        RZ(-1.5707963267948966, wires=[1])]

        """
        pi_half = qml.math.ones_like(phi) * (np.pi / 2)
        return [
            qml.RZ(pi_half, wires=wires[1]),
            qml.RY(phi / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
            qml.RY(-phi / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
            qml.RZ(-pi_half, wires=wires[1]),
        ]


class CRY(ControlledOp):
    r"""The controlled-RY operator

    .. math::

        \begin{align}
            CR_y(\phi) &=
            \begin{bmatrix}
                1 & 0 & 0 & 0 \\
                0 & 1 & 0 & 0\\
                0 & 0 & \cos(\phi/2) & -\sin(\phi/2)\\
                0 & 0 & \sin(\phi/2) & \cos(\phi/2)
            \end{bmatrix}.
        \end{align}

    **Details:**

    * Number of wires: 2
    * Number of parameters: 1
    * Number of dimensions per parameter: (0,)
    * Gradient recipe: The controlled-RY operator satisfies a four-term parameter-shift rule
      (see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):

      .. math::

          \frac{d}{d\phi}f(CR_y(\phi)) = c_+ \left[f(CR_y(\phi+a)) - f(CR_y(\phi-a))\right] - c_- \left[f(CR_y(\phi+b)) - f(CR_y(\phi-b))\right]

      where :math:`f` is an expectation value depending on :math:`CR_y(\phi)`, and

      - :math:`a = \pi/2`
      - :math:`b = 3\pi/2`
      - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`

    Args:
        phi (float): rotation angle :math:`\phi`
        wires (Sequence[int]): the wire the operation acts on
        id (str or None): String representing the operation (optional)
    """

    num_wires = 2
    """int: Number of wires that the operation acts on."""

    num_params = 1
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = (0,)
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CRY"
    parameter_frequencies = [(0.5, 1.0)]

    def __init__(self, phi, wires, id=None):
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.RY, phi, wires=wires[1:])
        super().__init__(base, control_wires=wires[:1], id=id)

    def __repr__(self):
        return f"CRY({self.data[0]}, wires={self.wires.tolist()}))"

    def _flatten(self):
        return self.data, (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(*data, wires=metadata[0])

    @classmethod
    def _primitive_bind_call(cls, phi, wires, id=None):
        return cls._primitive.bind(phi, *wires, n_wires=len(wires))

    @staticmethod
    def compute_matrix(theta):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CRY.matrix`


        Args:
            theta (tensor_like or float): rotation angle

        Returns:
            tensor_like: canonical matrix

        **Example**

        >>> qml.CRY.compute_matrix(torch.tensor(0.5))
        tensor([[ 1.0000+0.j,  0.0000+0.j,  0.0000+0.j,  0.0000+0.j],
                [ 0.0000+0.j,  1.0000+0.j,  0.0000+0.j,  0.0000+0.j],
                [ 0.0000+0.j,  0.0000+0.j,  0.9689+0.j, -0.2474-0.j],
                [ 0.0000+0.j,  0.0000+0.j,  0.2474+0.j,  0.9689+0.j]])
        """
        interface = qml.math.get_interface(theta)

        c = qml.math.cos(theta / 2)
        s = qml.math.sin(theta / 2)

        if interface == "tensorflow":
            c = qml.math.cast_like(c, 1j)
            s = qml.math.cast_like(s, 1j)

        # The following avoids casting an imaginary quantity to reals when back propagating
        c = (1 + 0j) * c
        s = (1 + 0j) * s
        ones = qml.math.ones_like(s)
        zeros = qml.math.zeros_like(s)
        matrix = [
            [ones, zeros, zeros, zeros],
            [zeros, ones, zeros, zeros],
            [zeros, zeros, c, -s],
            [zeros, zeros, s, c],
        ]

        return qml.math.stack([stack_last(row) for row in matrix], axis=-2)

    @staticmethod
    def compute_decomposition(phi, wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method). :

        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.CRY.decomposition`.

        Args:
            phi (float): rotation angle :math:`\phi`
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> qml.CRY.compute_decomposition(1.2, wires=(0,1))
        [RY(0.6, wires=[1]),
        CNOT(wires=[0, 1]),
        RY(-0.6, wires=[1]),
        CNOT(wires=[0, 1])]

        """
        return [
            qml.RY(phi / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
            qml.RY(-phi / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
        ]


class CRZ(ControlledOp):
    r"""The controlled-RZ operator

    .. math::

        \begin{align}
             CR_z(\phi) &=
             \begin{bmatrix}
                1 & 0 & 0 & 0 \\
                0 & 1 & 0 & 0\\
                0 & 0 & e^{-i\phi/2} & 0\\
                0 & 0 & 0 & e^{i\phi/2}
            \end{bmatrix}.
        \end{align}


    .. note:: The subscripts of the operations in the formula refer to the wires they act on, e.g. 1 corresponds
        to the first element in ``wires`` that is the **control qubit**.

    **Details:**

    * Number of wires: 2
    * Number of parameters: 1
    * Number of dimensions per parameter: (0,)
    * Gradient recipe: The controlled-RZ operator satisfies a four-term parameter-shift rule
      (see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):

      .. math::

          \frac{d}{d\phi}f(CR_z(\phi)) = c_+ \left[f(CR_z(\phi+a)) - f(CR_z(\phi-a))\right] - c_- \left[f(CR_z(\phi+b)) - f(CR_z(\phi-b))\right]

      where :math:`f` is an expectation value depending on :math:`CR_z(\phi)`, and

      - :math:`a = \pi/2`
      - :math:`b = 3\pi/2`
      - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`

    Args:
        phi (float): rotation angle :math:`\phi`
        wires (Sequence[int]): the wire the operation acts on
        id (str or None): String representing the operation (optional)

    """

    num_wires = 2
    """int: Number of wires that the operation acts on."""

    num_params = 1
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = (0,)
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CRZ"
    parameter_frequencies = [(0.5, 1.0)]

    def __init__(self, phi, wires, id=None):
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.RZ, phi, wires=wires[1:])
        super().__init__(base, control_wires=wires[:1], id=id)

    def __repr__(self):
        return f"CRZ({self.data[0]}, wires={self.wires})"

    def _flatten(self):
        return self.data, (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(*data, wires=metadata[0])

    @classmethod
    def _primitive_bind_call(cls, phi, wires, id=None):
        return cls._primitive.bind(phi, *wires, n_wires=len(wires))

    @staticmethod
    def compute_matrix(theta):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CRZ.matrix`

        Args:
            theta (tensor_like or float): rotation angle

        Returns:
            tensor_like: canonical matrix

        **Example**

        >>> qml.CRZ.compute_matrix(torch.tensor(0.5))
        tensor([[1.0+0.0j, 0.0+0.0j,       0.0+0.0j,       0.0+0.0j],
                [0.0+0.0j, 1.0+0.0j,       0.0+0.0j,       0.0+0.0j],
                [0.0+0.0j, 0.0+0.0j, 0.9689-0.2474j,       0.0+0.0j],
                [0.0+0.0j, 0.0+0.0j,       0.0+0.0j, 0.9689+0.2474j]])
        """
        if qml.math.get_interface(theta) == "tensorflow":
            p = qml.math.exp(-0.5j * qml.math.cast_like(theta, 1j))
            if qml.math.ndim(p) == 0:
                return qml.math.diag([1, 1, p, qml.math.conj(p)])

            ones = qml.math.ones_like(p)
            diags = stack_last([ones, ones, p, qml.math.conj(p)])
            return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)

        signs = qml.math.array([0, 0, 1, -1], like=theta)
        arg = -0.5j * theta

        if qml.math.ndim(arg) == 0:
            return qml.math.diag(qml.math.exp(arg * signs))

        diags = qml.math.exp(qml.math.outer(arg, signs))
        return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)

    @staticmethod
    def compute_eigvals(theta, **_):  # pylint: disable=arguments-differ
        r"""Eigenvalues of the operator in the computational basis (static method).

        If :attr:`diagonalizing_gates` are specified and implement a unitary :math:`U^{\dagger}`,
        the operator can be reconstructed as

        .. math:: O = U \Sigma U^{\dagger},

        where :math:`\Sigma` is the diagonal matrix containing the eigenvalues.

        Otherwise, no particular order for the eigenvalues is guaranteed.

        .. seealso:: :meth:`~.CRZ.eigvals`


        Args:
            theta (tensor_like or float): rotation angle

        Returns:
            tensor_like: eigenvalues

        **Example**

        >>> qml.CRZ.compute_eigvals(torch.tensor(0.5))
        tensor([1.0000+0.0000j, 1.0000+0.0000j, 0.9689-0.2474j, 0.9689+0.2474j])
        """
        if qml.math.get_interface(theta) == "tensorflow":
            phase = qml.math.exp(-0.5j * qml.math.cast_like(theta, 1j))
            ones = qml.math.ones_like(phase)
            return stack_last([ones, ones, phase, qml.math.conj(phase)])

        prefactors = qml.math.array([0, 0, -0.5j, 0.5j], like=theta)
        if qml.math.ndim(theta) == 0:
            product = theta * prefactors
        else:
            product = qml.math.outer(theta, prefactors)
        return qml.math.exp(product)

    def eigvals(self):
        return self.compute_eigvals(*self.parameters)

    @staticmethod
    def compute_decomposition(phi, wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method). :

        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.CRZ.decomposition`.

        Args:
            phi (float): rotation angle :math:`\phi`
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> qml.CRZ.compute_decomposition(1.2, wires=(0,1))
        [PhaseShift(0.6, wires=[1]),
        CNOT(wires=[0, 1]),
        PhaseShift(-0.6, wires=[1]),
        CNOT(wires=[0, 1])]

        """
        return [
            qml.PhaseShift(phi / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
            qml.PhaseShift(-phi / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
        ]


class CRot(ControlledOp):
    r"""The controlled-Rot operator

    .. math:: CR(\phi, \theta, \omega) = \begin{bmatrix}
            1 & 0 & 0 & 0 \\
            0 & 1 & 0 & 0\\
            0 & 0 & e^{-i(\phi+\omega)/2}\cos(\theta/2) & -e^{i(\phi-\omega)/2}\sin(\theta/2)\\
            0 & 0 & e^{-i(\phi-\omega)/2}\sin(\theta/2) & e^{i(\phi+\omega)/2}\cos(\theta/2)
        \end{bmatrix}.

    .. note:: The first wire provided corresponds to the **control qubit**.

    **Details:**

    * Number of wires: 2
    * Number of parameters: 3
    * Number of dimensions per parameter: (0, 0, 0)
    * Gradient recipe: The controlled-Rot operator satisfies a four-term parameter-shift rule
      (see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):

      .. math::

          \frac{d}{d\mathbf{x}_i}f(CR(\mathbf{x}_i)) = c_+ \left[f(CR(\mathbf{x}_i+a)) - f(CR(\mathbf{x}_i-a))\right] - c_- \left[f(CR(\mathbf{x}_i+b)) - f(CR(\mathbf{x}_i-b))\right]

      where :math:`f` is an expectation value depending on :math:`CR(\mathbf{x}_i)`, and

      - :math:`\mathbf{x} = (\phi, \theta, \omega)` and `i` is an index to :math:`\mathbf{x}`
      - :math:`a = \pi/2`
      - :math:`b = 3\pi/2`
      - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`

    Args:
        phi (float): rotation angle :math:`\phi`
        theta (float): rotation angle :math:`\theta`
        omega (float): rotation angle :math:`\omega`
        wires (Sequence[int]): the wire the operation acts on
        id (str or None): String representing the operation (optional)

    """

    num_wires = 2
    """int: Number of wires this operator acts on."""

    num_params = 3
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = (0, 0, 0)
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "CRot"
    parameter_frequencies = [(0.5, 1.0), (0.5, 1.0), (0.5, 1.0)]

    def __init__(
        self, phi, theta, omega, wires, id=None
    ):  # pylint: disable=too-many-arguments,too-many-positional-arguments
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.Rot, phi, theta, omega, wires=wires[1:])
        super().__init__(base, control_wires=wires[:1], id=id)

    def __repr__(self):
        params = ", ".join([repr(p) for p in self.parameters])
        return f"CRot({params}, wires={self.wires})"

    def _flatten(self):
        return self.data, (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(*data, wires=metadata[0])

    # pylint: disable=too-many-arguments
    @classmethod
    def _primitive_bind_call(
        cls, phi, theta, omega, wires, id=None
    ):  # pylint: disable=too-many-positional-arguments
        return cls._primitive.bind(phi, theta, omega, *wires, n_wires=len(wires))

    @staticmethod
    def compute_matrix(phi, theta, omega):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.CRot.matrix`


        Args:
            phi(tensor_like or float): first rotation angle
            theta (tensor_like or float): second rotation angle
            omega (tensor_like or float): third rotation angle

        Returns:
            tensor_like: canonical matrix

        **Example**

         >>> qml.CRot.compute_matrix(torch.tensor(0.1), torch.tensor(0.2), torch.tensor(0.3))
         tensor([[ 1.0+0.0j,  0.0+0.0j,        0.0+0.0j,        0.0+0.0j],
                [ 0.0+0.0j,  1.0+0.0j,        0.0+0.0j,        0.0+0.0j],
                [ 0.0+0.0j,  0.0+0.0j,  0.9752-0.1977j, -0.0993+0.0100j],
                [ 0.0+0.0j,  0.0+0.0j,  0.0993+0.0100j,  0.9752+0.1977j]])
        """
        # It might be that they are in different interfaces, e.g.,
        # CRot(0.2, 0.3, tf.Variable(0.5), wires=[0, 1])
        # So we need to make sure the matrix comes out having the right type
        interface = qml.math.get_interface(phi, theta, omega)

        c = qml.math.cos(theta / 2)
        s = qml.math.sin(theta / 2)

        # If anything is not tensorflow, it has to be casted
        if interface == "tensorflow":
            phi = qml.math.cast_like(qml.math.asarray(phi, like=interface), 1j)
            omega = qml.math.cast_like(qml.math.asarray(omega, like=interface), 1j)
            c = qml.math.cast_like(qml.math.asarray(c, like=interface), 1j)
            s = qml.math.cast_like(qml.math.asarray(s, like=interface), 1j)

        # The following variable is used to assert the all terms to be stacked have same shape
        one = qml.math.ones_like(phi) * qml.math.ones_like(omega)
        c = c * one
        s = s * one

        o = qml.math.ones_like(c)
        z = qml.math.zeros_like(c)
        mat = [
            [o, z, z, z],
            [z, o, z, z],
            [
                z,
                z,
                qml.math.exp(-0.5j * (phi + omega)) * c,
                -qml.math.exp(0.5j * (phi - omega)) * s,
            ],
            [
                z,
                z,
                qml.math.exp(-0.5j * (phi - omega)) * s,
                qml.math.exp(0.5j * (phi + omega)) * c,
            ],
        ]

        return qml.math.stack([stack_last(row) for row in mat], axis=-2)

    @staticmethod
    def compute_decomposition(phi, theta, omega, wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method). :

        .. math:: O = O_1 O_2 \dots O_n.


        .. seealso:: :meth:`~.CRot.decomposition`.

        Args:
            phi (float): rotation angle :math:`\phi`
            theta (float): rotation angle :math:`\theta`
            omega (float): rotation angle :math:`\omega`
            wires (Iterable, Wires): the wires the operation acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> qml.CRot.compute_decomposition(1.234, 2.34, 3.45, wires=[0, 1])
        [RZ(-1.108, wires=[1]),
         CNOT(wires=[0, 1]),
         RZ(-2.342, wires=[1]),
         RY(-1.17, wires=[1]),
         CNOT(wires=[0, 1]),
         RY(1.17, wires=[1]),
         RZ(3.45, wires=[1])]

        """
        return [
            qml.RZ((phi - omega) / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
            qml.RZ(-(phi + omega) / 2, wires=wires[1]),
            qml.RY(-theta / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
            qml.RY(theta / 2, wires=wires[1]),
            qml.RZ(omega, wires=wires[1]),
        ]


class ControlledPhaseShift(ControlledOp):
    r"""A qubit controlled phase shift.

    .. math:: CR_\phi(\phi) = \begin{bmatrix}
                1 & 0 & 0 & 0 \\
                0 & 1 & 0 & 0 \\
                0 & 0 & 1 & 0 \\
                0 & 0 & 0 & e^{i\phi}
            \end{bmatrix}.

    .. note:: The first wire provided corresponds to the **control qubit**.

    **Details:**

    * Number of wires: 2
    * Number of parameters: 1
    * Number of dimensions per parameter: (0,)
    * Gradient recipe: :math:`\frac{d}{d\phi}f(CR_\phi(\phi)) = \frac{1}{2}\left[f(CR_\phi(\phi+\pi/2)) - f(CR_\phi(\phi-\pi/2))\right]`
        where :math:`f` is an expectation value depending on :math:`CR_{\phi}(\phi)`.

    Args:
        phi (float): rotation angle :math:`\phi`
        wires (Sequence[int]): the wire the operation acts on
        id (str or None): String representing the operation (optional)

    """

    num_wires = 2
    """int: Number of wires the operator acts on."""

    num_params = 1
    """int: Number of trainable parameters that the operator depends on."""

    ndim_params = (0,)
    """tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""

    name = "ControlledPhaseShift"
    parameter_frequencies = [(1,)]

    def __init__(self, phi, wires, id=None):
        # We use type.__call__ instead of calling the class directly so that we don't bind the
        # operator primitive when new program capture is enabled
        base = type.__call__(qml.PhaseShift, phi, wires=wires[1:])
        super().__init__(base, control_wires=wires[:1], id=id)

    def __repr__(self):
        return f"ControlledPhaseShift({self.data[0]}, wires={self.wires})"

    def _flatten(self):
        return self.data, (self.wires,)

    @classmethod
    def _unflatten(cls, data, metadata):
        return cls(*data, wires=metadata[0])

    @classmethod
    def _primitive_bind_call(cls, phi, wires, id=None):
        return cls._primitive.bind(phi, *wires, n_wires=len(wires))

    @staticmethod
    def compute_matrix(phi):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a canonical matrix in the computational basis (static method).

        The canonical matrix is the textbook matrix representation that does not consider wires.
        Implicitly, this assumes that the wires of the operator correspond to the global wire order.

        .. seealso:: :meth:`~.ControlledPhaseShift.matrix`

        Args:
            phi (tensor_like or float): phase shift

        Returns:
            tensor_like: canonical matrix

        **Example**

        >>> qml.ControlledPhaseShift.compute_matrix(torch.tensor(0.5))
            tensor([[1.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.0000+0.0000j],
                    [0.0+0.0j, 1.0+0.0j, 0.0+0.0j, 0.0000+0.0000j],
                    [0.0+0.0j, 0.0+0.0j, 1.0+0.0j, 0.0000+0.0000j],
                    [0.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.8776+0.4794j]])
        """
        if qml.math.get_interface(phi) == "tensorflow":
            p = qml.math.exp(1j * qml.math.cast_like(phi, 1j))
            if qml.math.ndim(p) == 0:
                return qml.math.diag([1, 1, 1, p])

            ones = qml.math.ones_like(p)
            diags = stack_last([ones, ones, ones, p])
            return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)

        signs = qml.math.array([0, 0, 0, 1], like=phi)
        arg = 1j * phi

        if qml.math.ndim(arg) == 0:
            return qml.math.diag(qml.math.exp(arg * signs))

        diags = qml.math.exp(qml.math.outer(arg, signs))
        return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)

    @staticmethod
    def compute_eigvals(phi, **_):  # pylint: disable=arguments-differ
        r"""Eigenvalues of the operator in the computational basis (static method).

        If :attr:`diagonalizing_gates` are specified and implement a unitary :math:`U^{\dagger}`,
        the operator can be reconstructed as

        .. math:: O = U \Sigma U^{\dagger},

        where :math:`\Sigma` is the diagonal matrix containing the eigenvalues.

        Otherwise, no particular order for the eigenvalues is guaranteed.

        .. seealso:: :meth:`~.ControlledPhaseShift.eigvals`


        Args:
            phi (tensor_like or float): phase shift

        Returns:
            tensor_like: eigenvalues

        **Example**

        >>> qml.ControlledPhaseShift.compute_eigvals(torch.tensor(0.5))
        tensor([1.0000+0.0000j, 1.0000+0.0000j, 1.0000+0.0000j, 0.8776+0.4794j])
        """
        if qml.math.get_interface(phi) == "tensorflow":
            phase = qml.math.exp(1j * qml.math.cast_like(phi, 1j))
            ones = qml.math.ones_like(phase)
            return stack_last([ones, ones, ones, phase])

        prefactors = qml.math.array([0, 0, 0, 1j], like=phi)
        if qml.math.ndim(phi) == 0:
            product = phi * prefactors
        else:
            product = qml.math.outer(phi, prefactors)
        return qml.math.exp(product)

    def eigvals(self):
        return self.compute_eigvals(*self.parameters)

    @staticmethod
    def compute_decomposition(phi, wires):  # pylint: disable=arguments-differ
        r"""Representation of the operator as a product of other operators (static method). :

        .. math:: O = O_1 O_2 \dots O_n.

        .. seealso:: :meth:`~.ControlledPhaseShift.decomposition`.

        Args:
            phi (float): rotation angle :math:`\phi`
            wires (Iterable, Wires): wires that the operator acts on

        Returns:
            list[Operator]: decomposition into lower level operations

        **Example:**

        >>> qml.ControlledPhaseShift.compute_decomposition(1.234, wires=(0,1))
        [PhaseShift(0.617, wires=[0]),
         CNOT(wires=[0, 1]),
         PhaseShift(-0.617, wires=[1]),
         CNOT(wires=[0, 1]),
         PhaseShift(0.617, wires=[1])]

        """
        return [
            qml.PhaseShift(phi / 2, wires=wires[0]),
            qml.CNOT(wires=wires),
            qml.PhaseShift(-phi / 2, wires=wires[1]),
            qml.CNOT(wires=wires),
            qml.PhaseShift(phi / 2, wires=wires[1]),
        ]


CPhase = ControlledPhaseShift

_modules/pennylane/ops/op_math/controlled_ops

 

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