Source code for pennylane.templates.subroutines.qft

# Copyright 2018-2021 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 the template for QFT.
"""
# pylint:disable=abstract-method,arguments-differ,protected-access

import functools
import numpy as np

import pennylane as qml
from pennylane.operation import AnyWires, Operation


class QFT(Operation):
    r"""QFT(wires)
    Apply a quantum Fourier transform (QFT).

    For the :math:`N`-qubit computational basis state :math:`|m\rangle`, the QFT performs the
    transformation

    .. math::

        |m\rangle \rightarrow \frac{1}{\sqrt{2^{N}}}\sum_{n=0}^{2^{N} - 1}\omega_{N}^{mn} |n\rangle,

    where :math:`\omega_{N} = e^{\frac{2 \pi i}{2^{N}}}` is the :math:`2^{N}`-th root of unity.

    **Details:**

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

    Args:
        wires (int or Iterable[Number, str]]): the wire(s) the operation acts on

    **Example**

    The quantum Fourier transform is applied by specifying the corresponding wires:

    .. code-block::

        wires = 3

        dev = qml.device('default.qubit',wires=wires)

        @qml.qnode(dev)
        def circuit_qft(basis_state):
            qml.BasisState(basis_state, wires=range(wires))
            qml.QFT(wires=range(wires))
            return qml.state()

        circuit_qft(np.array([1.0, 0.0, 0.0], requires_grad=False))
    """
    num_wires = AnyWires
    grad_method = None

    def __init__(self, *params, wires=None, do_queue=True, id=None):
        wires = qml.wires.Wires(wires)
        self.hyperparameters["n_wires"] = len(wires)
        super().__init__(*params, wires=wires, do_queue=do_queue, id=id)

    @property
    def num_params(self):
        return 0

    @staticmethod
    @functools.lru_cache()
    def compute_matrix(n_wires):  # pylint: disable=arguments-differ
        dimension = 2**n_wires

        mat = np.zeros((dimension, dimension), dtype=np.complex128)
        omega = np.exp(2 * np.pi * 1j / dimension)

        for m in range(dimension):
            for n in range(dimension):
                mat[m, n] = omega ** (m * n)

        return mat / np.sqrt(dimension)

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

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


        .. seealso:: :meth:`~.QFT.decomposition`.

        Args:
            wires (Iterable, Wires): wires that the operator acts on
            n_wires (int): number of wires or ``len(wires)``

        Returns:
            list[Operator]: decomposition of the operator

        **Example:**

        >>> qml.QFT.compute_decomposition((0,1,2,4))
        [Toffoli(wires=[1, 2, 4]), CNOT(wires=[1, 2]), Toffoli(wires=[0, 2, 4])]

        """
        shifts = [2 * np.pi * 2**-i for i in range(2, n_wires + 1)]

        decomp_ops = []
        for i, wire in enumerate(wires):
            decomp_ops.append(qml.Hadamard(wire))

            for shift, control_wire in zip(shifts[: len(shifts) - i], wires[i + 1 :]):
                op = qml.ControlledPhaseShift(shift, wires=[control_wire, wire])
                decomp_ops.append(op)

        first_half_wires = wires[: n_wires // 2]
        last_half_wires = wires[-(n_wires // 2) :]

        for wire1, wire2 in zip(first_half_wires, reversed(last_half_wires)):
            swap = qml.SWAP(wires=[wire1, wire2])
            decomp_ops.append(swap)

        return decomp_ops

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