Source code for pennylane.ops.functions.eigvals

# Copyright 2018-2022 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 module contains the qml.eigvals function.
"""
import warnings

# pylint: disable=protected-access
from functools import reduce

import scipy

import pennylane as qml


@qml.op_transform
def eigvals(op, k=1, which="SA"):
    r"""The eigenvalues of one or more operations.

    .. note::

        For a :class:`~.SparseHamiltonian` object, the eigenvalues are computed with the efficient
        ``scipy.sparse.linalg.eigsh`` method which returns :math:`k` eigenvalues. The default value
        of :math:`k` is :math:`1`. For an :math:`N \times N` sparse matrix, :math:`k` must be
        smaller than :math:`N - 1`, otherwise ``scipy.sparse.linalg.eigsh`` fails. If the requested
        :math:`k` is equal or larger than :math:`N - 1`, the regular ``qml.math.linalg.eigvalsh``
        is applied on the dense matrix. For more details see the ``scipy.sparse.linalg.eigsh``
        `documentation <https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.eigsh.html#scipy.sparse.linalg.eigsh>`_.

    Args:
        op (.Operator, pennylane.QNode, .QuantumTape, or Callable): An operator, quantum node, tape,
            or function that applies quantum operations.
        k (int): The number of eigenvalues to be returned for a :class:`~.SparseHamiltonian`.
        which (str): Method for computing the eigenvalues of a :class:`~.SparseHamiltonian`. The
            possible methods are ``'LM'`` (largest in magnitude), ``'SM'`` (smallest in magnitude),
            ``'LA'`` (largest algebraic), ``'SA'`` (smallest algebraic) and ``'BE'`` (:math:`k/2`
            from each end of the spectrum).

    Returns:
        tensor_like or function: If an operator is provided as input, the eigenvalues are returned
        directly. If a QNode or quantum function is provided as input, a function which accepts the
        same arguments as the QNode or quantum function is returned. When called, this function will
        return the unitary matrix in the appropriate autodiff framework (Autograd, TensorFlow,
        PyTorch, JAX) given its parameters.

    **Example**

    Given an operation, ``qml.eigvals`` returns the eigenvalues:

    >>> op = qml.PauliZ(0) @ qml.PauliX(1) - 0.5 * qml.PauliY(1)
    >>> qml.eigvals(op)
    array([-1.11803399, -1.11803399,  1.11803399,  1.11803399])

    It can also be used in a functional form:

    >>> x = torch.tensor(0.6, requires_grad=True)
    >>> eigval_fn = qml.eigvals(qml.RX)
    >>> eigval_fn(x, wires=0)
    tensor([0.9553+0.2955j, 0.9553-0.2955j], grad_fn=<LinalgEigBackward>)

    In its functional form, it is fully differentiable with respect to gate arguments:

    >>> loss = torch.real(torch.sum(eigval_fn(x, wires=0)))
    >>> loss.backward()
    >>> x.grad
    tensor(-0.2955)

    This operator transform can also be applied to QNodes, tapes, and quantum functions
    that contain multiple operations; see Usage Details below for more details.

    .. details::
        :title: Usage Details

        ``qml.eigvals`` can also be used with QNodes, tapes, or quantum functions that
        contain multiple operations. However, in this situation, **eigenvalues may
        be computed numerically**. This can lead to a large computational overhead
        for a large number of wires.

        Consider the following quantum function:

        .. code-block:: python3

            def circuit(theta):
                qml.RX(theta, wires=1)
                qml.PauliZ(wires=0)

        We can use ``qml.eigvals`` to generate a new function that returns the eigenvalues
        corresponding to the function ``circuit``:

        >>> eigvals_fn = qml.eigvals(circuit)
        >>> theta = np.pi / 4
        >>> eigvals_fn(theta)
        array([ 0.92387953+0.38268343j,  0.92387953-0.38268343j,
               -0.92387953+0.38268343j, -0.92387953-0.38268343j])
    """
    if isinstance(op, qml.Hamiltonian):
        warnings.warn(
            "For Hamiltonians, the eigenvalues will be computed numerically. "
            "This may be computationally intensive for a large number of wires. "
            "Consider using a sparse representation of the Hamiltonian with qml.SparseHamiltonian.",
            UserWarning,
        )
        return qml.math.linalg.eigvalsh(qml.matrix(op))

    if isinstance(op, qml.SparseHamiltonian):
        sparse_matrix = op.sparse_matrix()
        if k < sparse_matrix.shape[0] - 1:
            return scipy.sparse.linalg.eigsh(sparse_matrix, k=k, which=which)[0]
        return qml.math.linalg.eigvalsh(sparse_matrix.toarray())

    # TODO: make `eigvals` take a `wire_order` argument to mimic `matrix`
    return op.eigvals()


@eigvals.tape_transform
def _eigvals(tape):
    op_wires = [op.wires for op in tape.operations]
    all_wires = qml.wires.Wires.all_wires(op_wires).tolist()
    unique_wires = qml.wires.Wires.unique_wires(op_wires).tolist()

    if len(all_wires) != len(unique_wires):
        warnings.warn(
            "For multiple operations, the eigenvalues will be computed numerically. "
            "This may be computationally intensive for a large number of wires.",
            UserWarning,
        )
        return qml.math.linalg.eigvals(qml.matrix(tape))

    # TODO: take into account wire ordering, by reordering eigenvalues
    # as per operator wires/wire ordering, and by inserting implicit identity
    # matrices (eigenvalues [1, 1]) at missing locations.

    ev = [eigvals(op) for op in tape.operations]

    if len(ev) == 1:
        return ev[0]

    return reduce(qml.math.kron, ev)

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