Source code for pennylane.templates.subroutines.grover

# 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.
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"""
Contains the Grover Operation template.
"""
import itertools
import functools
import numpy as np
from pennylane.operation import AnyWires, Operation
from pennylane.ops import Hadamard, PauliZ, MultiControlledX


class GroverOperator(Operation):
    r"""Performs the Grover Diffusion Operator.

    .. math::

        G = 2 |s \rangle \langle s | - I
        = H^{\bigotimes n} \left( 2 |0\rangle \langle 0| - I \right) H^{\bigotimes n}

    where :math:`n` is the number of wires, and :math:`|s\rangle` is the uniform superposition:

    .. math::

        |s\rangle = H^{\bigotimes n} |0\rangle =  \frac{1}{\sqrt{2^n}} \sum_{i=0}^{2^n-1} | i \rangle.

    For this template, the operator is implemented with a layer of Hadamards, a layer of :math:`X`,
    followed by a multi-controlled :math:`Z` gate, then another layer of :math:`X` and Hadamards.
    This is expressed in a compact form by the circuit below:

    .. figure:: ../../_static/templates/subroutines/grover.svg
        :align: center
        :width: 60%
        :target: javascript:void(0);

    The open circles on the controlled gate indicate control on 0 instead of 1.
    The ``Z`` gates on the last wire result from leveraging the circuit identity :math:`HXH = Z`,
    where the last ``H`` gate converts the multi-controlled :math:`Z` gate into a
    multi-controlled :math:`X` gate.

    Args:
        wires (Union[Wires, Sequence[int], or int]): the wires to apply to
        work_wires (Union[Wires, Sequence[int], or int]): optional auxiliary wires to assist
            in the decomposition of :class:`~.MultiControlledX`.

    **Example**

    The Grover Diffusion Operator amplifies the magnitude of the basis state with
    a negative phase.  For example, if the solution to the search problem is the :math:`|111\rangle`
    state, we require an oracle that flips its phase; this could be implemented using a `CCZ` gate:

    .. code-block:: python

        n_wires = 3
        wires = list(range(n_wires))

        def oracle():
            qml.Hadamard(wires[-1])
            qml.Toffoli(wires=wires)
            qml.Hadamard(wires[-1])

    We can then implement the entire Grover Search Algorithm for ``num_iterations`` iterations by alternating calls to the oracle and the diffusion operator:

    .. code-block:: python

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

        @qml.qnode(dev)
        def GroverSearch(num_iterations=1):
            for wire in wires:
                qml.Hadamard(wire)

            for _ in range(num_iterations):
                oracle()
                qml.templates.GroverOperator(wires=wires)
            return qml.probs(wires)

    >>> GroverSearch(num_iterations=1)
    tensor([0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125,
            0.78125], requires_grad=True)
    >>> GroverSearch(num_iterations=2)
    tensor([0.0078125, 0.0078125, 0.0078125, 0.0078125, 0.0078125, 0.0078125,
        0.0078125, 0.9453125], requires_grad=True)

    We can see that the marked :math:`|111\rangle` state has the greatest probability amplitude.

    Optimally, the oracle-operator pairing should be repeated :math:`\lceil \frac{\pi}{4}\sqrt{2^{n}} \rceil` times.

    """
    num_wires = AnyWires
    grad_method = None

    def __init__(self, wires=None, work_wires=None, do_queue=True, id=None):
        if (not hasattr(wires, "__len__")) or (len(wires) < 2):
            raise ValueError("GroverOperator must have at least two wires provided.")

        self._hyperparameters = {"n_wires": len(wires), "work_wires": work_wires}

        super().__init__(wires=wires, do_queue=do_queue, id=id)

    @property
    def num_params(self):
        return 0

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

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



        .. seealso:: :meth:`~.GroverOperator.decomposition`.

        Args:
            wires (Any or Iterable[Any]): wires that the operator acts on
            work_wires (Any or Iterable[Any]): optional auxiliary wires to assist
                in the decomposition of :class:`~.MultiControlledX`.

        Returns:
            list[.Operator]: decomposition of the operator
        """
        ctrl_str = "0" * (len(wires) - 1)

        op_list = []

        for wire in wires[:-1]:
            op_list.append(Hadamard(wire))

        op_list.append(PauliZ(wires[-1]))
        op_list.append(
            MultiControlledX(
                control_values=ctrl_str,
                wires=wires,
                work_wires=work_wires,
            )
        )

        op_list.append(PauliZ(wires[-1]))

        for wire in wires[:-1]:
            op_list.append(Hadamard(wire))

        return op_list

    @staticmethod
    @functools.lru_cache()
    def compute_matrix(n_wires, work_wires):  # pylint: disable=arguments-differ,unused-argument
        # s1 = H|0>, Hadamard on a single qubit in the ground state
        s1 = np.array([1, 1]) / np.sqrt(2)

        # uniform superposition state |s>
        s = functools.reduce(np.kron, list(itertools.repeat(s1, n_wires)))

        # Grover diffusion operator
        G = 2 * np.outer(s, s) - np.identity(2**n_wires)
        return G

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