# 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.

"""
A collection of useful quantum information functions for operators.
"""

import logging
import warnings
import numpy as np
from scipy import sparse

from qiskit.exceptions import QiskitError, MissingOptionalLibraryError
from qiskit.circuit.gate import Gate
from qiskit.quantum_info.operators.base_operator import BaseOperator
from qiskit.quantum_info.operators.operator import Operator
from qiskit.quantum_info.operators.channel.quantum_channel import QuantumChannel
from qiskit.quantum_info.operators.channel import Choi, SuperOp
from qiskit.quantum_info.states.densitymatrix import DensityMatrix
from qiskit.quantum_info.states.measures import state_fidelity

try:
    import cvxpy

    _HAS_CVX = True
except ImportError:
    _HAS_CVX = False

logger = logging.getLogger(__name__)


def process_fidelity(channel, target=None, require_cp=True, require_tp=True):
    r"""Return the process fidelity of a noisy quantum channel.


    The process fidelity :math:`F_{\text{pro}}(\mathcal{E}, \mathcal{F})`
    between two quantum channels :math:`\mathcal{E}, \mathcal{F}` is given by

    .. math::
        F_{\text{pro}}(\mathcal{E}, \mathcal{F})
            = F(\rho_{\mathcal{E}}, \rho_{\mathcal{F}})

    where :math:`F` is the :func:`~qiskit.quantum_info.state_fidelity`,
    :math:`\rho_{\mathcal{E}} = \Lambda_{\mathcal{E}} / d` is the
    normalized :class:`~qiskit.quantum_info.Choi` matrix for the channel
    :math:`\mathcal{E}`, and :math:`d` is the input dimension of
    :math:`\mathcal{E}`.

    When the target channel is unitary this is equivalent to

    .. math::
        F_{\text{pro}}(\mathcal{E}, U)
            = \frac{Tr[S_U^\dagger S_{\mathcal{E}}]}{d^2}

    where :math:`S_{\mathcal{E}}, S_{U}` are the
    :class:`~qiskit.quantum_info.SuperOp` matrices for the *input* quantum
    channel :math:`\mathcal{E}` and *target* unitary :math:`U` respectively,
    and :math:`d` is the input dimension of the channel.

    Args:
        channel (Operator or QuantumChannel): input quantum channel.
        target (Operator or QuantumChannel or None): target quantum channel.
            If `None` target is the identity operator [Default: None].
        require_cp (bool): check if input and target channels are
                           completely-positive and if non-CP log warning
                           containing negative eigenvalues of Choi-matrix
                           [Default: True].
        require_tp (bool): check if input and target channels are
                           trace-preserving and if non-TP log warning
                           containing negative eigenvalues of partial
                           Choi-matrix :math:`Tr_{\mbox{out}}[\mathcal{E}] - I`
                           [Default: True].

    Returns:
        float: The process fidelity :math:`F_{\text{pro}}`.

    Raises:
        QiskitError: if the channel and target do not have the same dimensions.
    """
    # Format inputs
    channel = _input_formatter(channel, SuperOp, "process_fidelity", "channel")
    target = _input_formatter(target, Operator, "process_fidelity", "target")

    if target:
        # Validate dimensions
        if channel.dim != target.dim:
            raise QiskitError(
                "Input quantum channel and target unitary must have the same "
                "dimensions ({} != {}).".format(channel.dim, target.dim)
            )

    # Validate complete-positivity and trace-preserving
    for label, chan in [("Input", channel), ("Target", target)]:
        if chan is not None and require_cp:
            cp_cond = _cp_condition(chan)
            neg = cp_cond < -1 * chan.atol
            if np.any(neg):
                logger.warning(
                    "%s channel is not CP. Choi-matrix has negative eigenvalues: %s",
                    label,
                    cp_cond[neg],
                )
        if chan is not None and require_tp:
            tp_cond = _tp_condition(chan)
            non_zero = np.logical_not(np.isclose(tp_cond, 0, atol=chan.atol, rtol=chan.rtol))
            if np.any(non_zero):
                logger.warning(
                    "%s channel is not TP. Tr_2[Choi] - I has non-zero eigenvalues: %s",
                    label,
                    tp_cond[non_zero],
                )

    if isinstance(target, Operator):
        # Compute fidelity with unitary target by applying the inverse
        # to channel and computing fidelity with the identity
        channel = channel.compose(target.adjoint())
        target = None

    input_dim, _ = channel.dim
    if target is None:
        # Compute process fidelity with identity channel
        if isinstance(channel, Operator):
            # |Tr[U]/dim| ** 2
            fid = np.abs(np.trace(channel.data) / input_dim) ** 2
        else:
            # Tr[S] / (dim ** 2)
            fid = np.trace(SuperOp(channel).data) / (input_dim ** 2)
        return float(np.real(fid))

    # For comparing two non-unitary channels we compute the state fidelity of
    # the normalized Choi-matrices. This is equivalent to the previous definition
    # when the target is a unitary channel.
    state1 = DensityMatrix(Choi(channel).data / input_dim)
    state2 = DensityMatrix(Choi(target).data / input_dim)
    return state_fidelity(state1, state2, validate=False)


def average_gate_fidelity(channel, target=None, require_cp=True, require_tp=False):
    r"""Return the average gate fidelity of a noisy quantum channel.

    The average gate fidelity :math:`F_{\text{ave}}` is given by

    .. math::
        F_{\text{ave}}(\mathcal{E}, U)
            &= \int d\psi \langle\psi|U^\dagger
                \mathcal{E}(|\psi\rangle\!\langle\psi|)U|\psi\rangle \\
            &= \frac{d F_{\text{pro}}(\mathcal{E}, U) + 1}{d + 1}

    where :math:`F_{\text{pro}}(\mathcal{E}, U)` is the
    :meth:`~qiskit.quantum_info.process_fidelity` of the input quantum
    *channel* :math:`\mathcal{E}` with a *target* unitary :math:`U`, and
    :math:`d` is the dimension of the *channel*.

    Args:
        channel (QuantumChannel or Operator): noisy quantum channel.
        target (Operator or None): target unitary operator.
            If `None` target is the identity operator [Default: None].
        require_cp (bool): check if input and target channels are
                           completely-positive and if non-CP log warning
                           containing negative eigenvalues of Choi-matrix
                           [Default: True].
        require_tp (bool): check if input and target channels are
                           trace-preserving and if non-TP log warning
                           containing negative eigenvalues of partial
                           Choi-matrix :math:`Tr_{\mbox{out}}[\mathcal{E}] - I`
                           [Default: True].

    Returns:
        float: The average gate fidelity :math:`F_{\text{ave}}`.

    Raises:
        QiskitError: if the channel and target do not have the same dimensions,
                     or have different input and output dimensions.
    """
    # Format inputs
    channel = _input_formatter(channel, SuperOp, "average_gate_fidelity", "channel")
    target = _input_formatter(target, Operator, "average_gate_fidelity", "target")

    if target is not None:
        try:
            target = Operator(target)
        except QiskitError as ex:
            raise QiskitError(
                "Target channel is not a unitary channel. To compare "
                "two non-unitary channels use the "
                "`qiskit.quantum_info.process_fidelity` function instead."
            ) from ex
    dim, _ = channel.dim
    f_pro = process_fidelity(channel, target=target, require_cp=require_cp, require_tp=require_tp)
    return (dim * f_pro + 1) / (dim + 1)


def gate_error(channel, target=None, require_cp=True, require_tp=False):
    r"""Return the gate error of a noisy quantum channel.

    The gate error :math:`E` is given by the average gate infidelity

    .. math::
        E(\mathcal{E}, U) = 1 - F_{\text{ave}}(\mathcal{E}, U)

    where :math:`F_{\text{ave}}(\mathcal{E}, U)` is the
    :meth:`~qiskit.quantum_info.average_gate_fidelity` of the input
    quantum *channel* :math:`\mathcal{E}` with a *target* unitary
    :math:`U`.

    Args:
        channel (QuantumChannel): noisy quantum channel.
        target (Operator or None): target unitary operator.
            If `None` target is the identity operator [Default: None].
        require_cp (bool): check if input and target channels are
                           completely-positive and if non-CP log warning
                           containing negative eigenvalues of Choi-matrix
                           [Default: True].
        require_tp (bool): check if input and target channels are
                           trace-preserving and if non-TP log warning
                           containing negative eigenvalues of partial
                           Choi-matrix :math:`Tr_{\mbox{out}}[\mathcal{E}] - I`
                           [Default: True].

    Returns:
        float: The average gate error :math:`E`.

    Raises:
        QiskitError: if the channel and target do not have the same dimensions,
                     or have different input and output dimensions.
    """
    # Format inputs
    channel = _input_formatter(channel, SuperOp, "gate_error", "channel")
    target = _input_formatter(target, Operator, "gate_error", "target")
    return 1 - average_gate_fidelity(
        channel, target=target, require_cp=require_cp, require_tp=require_tp
    )


def diamond_norm(choi, **kwargs):
    r"""Return the diamond norm of the input quantum channel object.

    This function computes the completely-bounded trace-norm (often
    referred to as the diamond-norm) of the input quantum channel object
    using the semidefinite-program from reference [1].

    Args:
        choi(Choi or QuantumChannel): a quantum channel object or
                                      Choi-matrix array.
        kwargs: optional arguments to pass to CVXPY solver.

    Returns:
        float: The completely-bounded trace norm
               :math:`\|\mathcal{E}\|_{\diamond}`.

    Raises:
        QiskitError: if CVXPY package cannot be found.

    Additional Information:
        The input to this function is typically *not* a CPTP quantum
        channel, but rather the *difference* between two quantum channels
        :math:`\|\Delta\mathcal{E}\|_\diamond` where
        :math:`\Delta\mathcal{E} = \mathcal{E}_1 - \mathcal{E}_2`.

    Reference:
        J. Watrous. "Simpler semidefinite programs for completely bounded
        norms", arXiv:1207.5726 [quant-ph] (2012).

    .. note::

        This function requires the optional CVXPY package to be installed.
        Any additional kwargs will be passed to the ``cvxpy.solve``
        function. See the CVXPY documentation for information on available
        SDP solvers.
    """
    _cvxpy_check("`diamond_norm`")  # Check CVXPY is installed

    choi = Choi(_input_formatter(choi, Choi, "diamond_norm", "choi"))

    def cvx_bmat(mat_r, mat_i):
        """Block matrix for embedding complex matrix in reals"""
        return cvxpy.bmat([[mat_r, -mat_i], [mat_i, mat_r]])

    # Dimension of input and output spaces
    dim_in = choi._input_dim
    dim_out = choi._output_dim
    size = dim_in * dim_out

    # SDP Variables to convert to real valued problem
    r0_r = cvxpy.Variable((dim_in, dim_in))
    r0_i = cvxpy.Variable((dim_in, dim_in))
    r0 = cvx_bmat(r0_r, r0_i)

    r1_r = cvxpy.Variable((dim_in, dim_in))
    r1_i = cvxpy.Variable((dim_in, dim_in))
    r1 = cvx_bmat(r1_r, r1_i)

    x_r = cvxpy.Variable((size, size))
    x_i = cvxpy.Variable((size, size))
    iden = sparse.eye(dim_out)

    # Watrous uses row-vec convention for his Choi matrix while we use
    # col-vec. It turns out row-vec convention is requried for CVXPY too
    # since the cvxpy.kron function must have a constant as its first argument.
    c_r = cvxpy.bmat([[cvxpy.kron(iden, r0_r), x_r], [x_r.T, cvxpy.kron(iden, r1_r)]])
    c_i = cvxpy.bmat([[cvxpy.kron(iden, r0_i), x_i], [-x_i.T, cvxpy.kron(iden, r1_i)]])
    c = cvx_bmat(c_r, c_i)

    # Convert col-vec convention Choi-matrix to row-vec convention and
    # then take Transpose: Choi_C -> Choi_R.T
    choi_rt = np.transpose(
        np.reshape(choi.data, (dim_in, dim_out, dim_in, dim_out)), (3, 2, 1, 0)
    ).reshape(choi.data.shape)
    choi_rt_r = choi_rt.real
    choi_rt_i = choi_rt.imag

    # Constraints
    cons = [
        r0 >> 0,
        r0_r == r0_r.T,
        r0_i == -r0_i.T,
        cvxpy.trace(r0_r) == 1,
        r1 >> 0,
        r1_r == r1_r.T,
        r1_i == -r1_i.T,
        cvxpy.trace(r1_r) == 1,
        c >> 0,
    ]

    # Objective function
    obj = cvxpy.Maximize(cvxpy.trace(choi_rt_r @ x_r) + cvxpy.trace(choi_rt_i @ x_i))
    prob = cvxpy.Problem(obj, cons)
    sol = prob.solve(**kwargs)
    return sol


def _cvxpy_check(name):
    """Check that a supported CVXPY version is installed"""
    # Check if CVXPY package is installed
    if not _HAS_CVX:
        raise QiskitError(
            "CVXPY backage is requried for {}. Install"
            " with `pip install cvxpy` to use.".format(name)
        )
    # Check CVXPY version
    version = cvxpy.__version__
    if version[0] != "1":
        raise MissingOptionalLibraryError(
            "CVXPY >= 1.0",
            "diamond_norm",
            msg=f"Incompatible CVXPY version {version} found.",
        )


# pylint: disable=too-many-return-statements
def _input_formatter(obj, fallback_class, func_name, arg_name):
    """Formatting function for input conversion"""
    # Empty input
    if obj is None:
        return obj

    # Channel-like input
    if isinstance(obj, QuantumChannel):
        return obj
    if hasattr(obj, "to_quantumchannel"):
        return obj.to_quantumchannel()
    if hasattr(obj, "to_channel"):
        return obj.to_channel()

    # Unitary-like input
    if isinstance(obj, (Gate, BaseOperator)):
        return Operator(obj)
    if hasattr(obj, "to_operator"):
        return obj.to_operator()

    warnings.warn(
        "Passing in a list or Numpy array to `{}` `{}` argument is "
        "deprecated as of 0.17.0 since the matrix representation cannot be inferred "
        "unambiguously. Use a Gate or BaseOperator subclass (eg. Operator, "
        "SuperOp, Choi) object instead.".format(func_name, arg_name),
        DeprecationWarning,
    )
    warnings.warn(f"Treating array input as a {fallback_class.__name__} object")
    return fallback_class(obj)


def _cp_condition(channel):
    """Return Choi-matrix eigenvalues for checking if channel is CP"""
    if isinstance(channel, QuantumChannel):
        if not isinstance(channel, Choi):
            channel = Choi(channel)
        return np.linalg.eigvalsh(channel.data)
    unitary = Operator(channel).data
    return np.tensordot(unitary, unitary.conj(), axes=([0, 1], [0, 1])).real


def _tp_condition(channel):
    """Return partial tr Choi-matrix eigenvalues for checking if channel is TP"""
    if isinstance(channel, QuantumChannel):
        if not isinstance(channel, Choi):
            channel = Choi(channel)
        choi = channel.data
        dims = tuple(np.sqrt(choi.shape).astype(int))
        shape = dims + dims
        tr_choi = np.trace(np.reshape(choi, shape), axis1=1, axis2=3)
    else:
        unitary = Operator(channel).data
        tr_choi = np.tensordot(unitary, unitary.conj(), axes=(0, 0))
    return np.linalg.eigvalsh(tr_choi - np.eye(len(tr_choi)))
