# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020.
#
# 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.

"""Algebra utilities and the ``GateSequence`` class."""

from __future__ import annotations

from collections.abc import Sequence
import math
import numpy as np

from qiskit.circuit import Gate, QuantumCircuit, Qubit
from qiskit.extensions import UnitaryGate


class GateSequence:
    """A class implementing a sequence of gates.

    This class stores the sequence of gates along with the unitary they implement.
    """

    def __init__(self, gates: Sequence[Gate] = ()) -> None:
        """Create a new sequence of gates.

        Args:
            gates: The gates in the sequence. The default is [].
        """
        self.gates = list(gates)
        self.matrices = [np.asarray(gate, dtype=np.complex128) for gate in gates]
        self.labels = [gate.name for gate in gates]

        # get U(2) representation of the gate sequence
        u2_matrix = np.identity(2)
        for matrix in self.matrices:
            # idea: could this be optimized by a specific numpy operation?
            u2_matrix = matrix.dot(u2_matrix)

        # convert to SU(2)
        su2_matrix, global_phase = _convert_u2_to_su2(u2_matrix)

        # convert to SO(3), that's what the Solovay Kitaev algorithm uses
        so3_matrix = _convert_su2_to_so3(su2_matrix)

        # store the matrix and the global phase
        self._eulers = None
        self.name = " ".join(self.labels)
        self.global_phase = global_phase
        self.product = so3_matrix
        self.product_su2 = su2_matrix

    def remove_cancelling_pair(self, indices: Sequence[int]) -> None:
        """Remove a pair of indices that cancel each other and *do not* change the matrices."""
        for index in list(indices[::-1]):
            self.gates.pop(index)
            self.labels.pop(index)

        # restore name
        self.name = " ".join(self.labels)

    def __eq__(self, other: "GateSequence") -> bool:
        """Check if this GateSequence is the same as the other GateSequence.

        Args:
            other: The GateSequence that will be compared to ``self``.

        Returns:
            True if ``other`` is equivalent to ``self``, false otherwise.

        """
        if not len(self.gates) == len(other.gates):
            return False

        for gate1, gate2 in zip(self.gates, other.gates):
            if gate1 != gate2:
                return False

        if self.global_phase != other.global_phase:
            return False

        return True

    def to_circuit(self):
        """Convert to a circuit.

        If no gates set but the product is not the identity, returns a circuit with a
        unitary operation to implement the matrix.
        """
        if len(self.gates) == 0 and not np.allclose(self.product, np.identity(3)):
            circuit = QuantumCircuit(1, global_phase=self.global_phase)
            su2 = _convert_so3_to_su2(self.product)
            circuit.unitary(su2, [0])
            return circuit

        circuit = QuantumCircuit(1, global_phase=self.global_phase)
        for gate in self.gates:
            circuit.append(gate, [0])

        return circuit

    def to_dag(self):
        """Convert to a :class:`.DAGCircuit`.

        If no gates set but the product is not the identity, returns a circuit with a
        unitary operation to implement the matrix.
        """
        from qiskit.dagcircuit import DAGCircuit

        qreg = [Qubit()]
        dag = DAGCircuit()
        dag.add_qubits(qreg)

        if len(self.gates) == 0 and not np.allclose(self.product, np.identity(3)):
            su2 = _convert_so3_to_su2(self.product)
            dag.apply_operation_back(UnitaryGate(su2), qreg)
            return dag

        dag.global_phase = self.global_phase
        for gate in self.gates:
            dag.apply_operation_back(gate, qreg)

        return dag

    def append(self, gate: Gate) -> "GateSequence":
        """Append gate to the sequence of gates.

        Args:
            gate: The gate to be appended.

        Returns:
            GateSequence with ``gate`` appended.
        """
        # invalidate euler angles and name
        self._eulers = None

        # TODO: this recomputes the product whenever we append something, which could be more
        # efficient by storing the current matrix and just multiplying the input gate to it
        # self.product = convert_su2_to_so3(self._compute_product(self.gates))
        matrix = np.array(gate, dtype=np.complex128)
        su2, phase = _convert_u2_to_su2(matrix)
        so3 = _convert_su2_to_so3(su2)

        self.product = so3.dot(self.product)
        self.product_su2 = su2.dot(self.product_su2)
        self.global_phase = self.global_phase + phase

        self.gates.append(gate)
        if len(self.labels) > 0:
            self.name += f" {gate.name}"
        else:
            self.name = gate.name
        self.labels.append(gate.name)

        self.matrices.append(matrix)

        return self

    def adjoint(self) -> "GateSequence":
        """Get the complex conjugate."""
        # We're initializing an empty GateSequence and set the state manually, as we can
        # efficiently infer the adjoint values from the current value instead of recomputing them.
        adjoint = GateSequence()
        adjoint.gates = [gate.inverse() for gate in reversed(self.gates)]
        adjoint.labels = [inv.name for inv in adjoint.gates]
        adjoint.name = " ".join(adjoint.labels)
        adjoint.product = np.conj(self.product).T
        adjoint.product_su2 = np.conj(self.product_su2).T
        adjoint.global_phase = -self.global_phase

        return adjoint

    def copy(self) -> "GateSequence":
        """Create copy of the sequence of gates.

        Returns:
            A new ``GateSequence`` containing copy of list of gates.

        """
        out = type(self).__new__(type(self))
        out.labels = self.labels.copy()
        out.gates = self.gates.copy()
        out.matrices = self.matrices.copy()
        out.global_phase = self.global_phase
        out.product = self.product.copy()
        out.product_su2 = self.product_su2.copy()
        out.name = self.name
        out._eulers = self._eulers
        return out

    def __len__(self) -> int:
        """Return length of sequence of gates.

        Returns:
            Length of list containing gates.
        """
        return len(self.gates)

    def __getitem__(self, index: int) -> Gate:
        """Returns the gate at ``index`` from the list of gates.

        Args
            index: Index of gate in list that will be returned.

        Returns:
            The gate at ``index`` in the list of gates.
        """
        return self.gates[index]

    def __repr__(self) -> str:
        """Return string representation of this object.

        Returns:
            Representation of this sequence of gates.
        """
        out = "["
        for gate in self.gates:
            out += gate.name
            out += ", "
        out += "]"
        out += ", product: "
        out += str(self.product)
        return out

    def __str__(self) -> str:
        """Return string representation of this object.

        Returns:
            Representation of this sequence of gates.
        """
        out = "["
        for gate in self.gates:
            out += gate.name
            out += ", "
        out += "]"
        out += ", product: \n"
        out += str(self.product)
        return out

    @classmethod
    def from_matrix(cls, matrix: np.ndarray) -> "GateSequence":
        """Initialize the gate sequence from a matrix, without a gate sequence.

        Args:
            matrix: The matrix, can be SU(2) or SO(3).

        Returns:
            A ``GateSequence`` initialized from the input matrix.

        Raises:
            ValueError: If the matrix has an invalid shape.
        """
        instance = cls()
        if matrix.shape == (2, 2):
            instance.product = _convert_su2_to_so3(matrix)
        elif matrix.shape == (3, 3):
            instance.product = matrix
        else:
            raise ValueError(f"Matrix must have shape (3, 3) or (2, 2) but has {matrix.shape}.")

        instance.gates = []
        return instance

    def dot(self, other: "GateSequence") -> "GateSequence":
        """Compute the dot-product with another gate sequence.

        Args:
            other: The other gate sequence.

        Returns:
            The dot-product as gate sequence.
        """
        # We're initializing an empty GateSequence and set the state manually, as we can more
        # efficiently compute the multiplied values from the already constructed matrices.
        composed = GateSequence()
        composed.gates = other.gates + self.gates
        composed.labels = other.labels + self.labels
        composed.name = " ".join(composed.labels)
        composed.product = np.dot(self.product, other.product)
        composed.global_phase = self.global_phase + other.global_phase

        return composed


def _convert_u2_to_su2(u2_matrix: np.ndarray) -> tuple[np.ndarray, float]:
    """Convert a U(2) matrix to SU(2) by adding a global phase."""
    z = 1 / np.sqrt(np.linalg.det(u2_matrix))
    su2_matrix = z * u2_matrix
    phase = np.arctan2(np.imag(z), np.real(z))

    return su2_matrix, phase


def _compute_euler_angles_from_so3(matrix: np.ndarray) -> tuple[float, float, float]:
    """Computes the Euler angles from the SO(3)-matrix u.

    Uses the algorithm from Gregory Slabaugh,
    see `here <https://www.gregslabaugh.net/publications/euler.pdf>`_.

    Args:
        matrix: The SO(3)-matrix for which the Euler angles need to be computed.

    Returns:
        Tuple (phi, theta, psi), where phi is rotation about z-axis, theta rotation about y-axis
        and psi rotation about x-axis.
    """
    matrix = np.round(matrix, decimals=10)
    if matrix[2][0] != 1 and matrix[2][1] != -1:
        theta = -math.asin(matrix[2][0])
        psi = math.atan2(matrix[2][1] / math.cos(theta), matrix[2][2] / math.cos(theta))
        phi = math.atan2(matrix[1][0] / math.cos(theta), matrix[0][0] / math.cos(theta))
        return phi, theta, psi
    else:
        phi = 0
        if matrix[2][0] == 1:
            theta = math.pi / 2
            psi = phi + math.atan2(matrix[0][1], matrix[0][2])
        else:
            theta = -math.pi / 2
            psi = -phi + math.atan2(-matrix[0][1], -matrix[0][2])
        return phi, theta, psi


def _compute_su2_from_euler_angles(angles: tuple[float, float, float]) -> np.ndarray:
    """Computes SU(2)-matrix from Euler angles.

    Args:
        angles: The tuple containing the Euler angles for which the corresponding SU(2)-matrix
            needs to be computed.

    Returns:
        The SU(2)-matrix corresponding to the Euler angles in angles.
    """
    phi, theta, psi = angles
    uz_phi = np.array([[np.exp(-0.5j * phi), 0], [0, np.exp(0.5j * phi)]], dtype=complex)
    uy_theta = np.array(
        [[math.cos(theta / 2), math.sin(theta / 2)], [-math.sin(theta / 2), math.cos(theta / 2)]],
        dtype=complex,
    )
    ux_psi = np.array(
        [[math.cos(psi / 2), math.sin(psi / 2) * 1j], [math.sin(psi / 2) * 1j, math.cos(psi / 2)]],
        dtype=complex,
    )
    return np.dot(uz_phi, np.dot(uy_theta, ux_psi))


def _convert_su2_to_so3(matrix: np.ndarray) -> np.ndarray:
    """Computes SO(3)-matrix from input SU(2)-matrix.

    Args:
        matrix: The SU(2)-matrix for which a corresponding SO(3)-matrix needs to be computed.

    Returns:
        The SO(3)-matrix corresponding to ``matrix``.

    Raises:
        ValueError: if ``matrix`` is not an SU(2)-matrix.
    """
    _check_is_su2(matrix)

    matrix = matrix.astype(complex)
    a = np.real(matrix[0][0])
    b = np.imag(matrix[0][0])
    c = -np.real(matrix[0][1])
    d = -np.imag(matrix[0][1])
    rotation = np.array(
        [
            [a**2 - b**2 - c**2 + d**2, 2 * a * b + 2 * c * d, -2 * a * c + 2 * b * d],
            [-2 * a * b + 2 * c * d, a**2 - b**2 + c**2 - d**2, 2 * a * d + 2 * b * c],
            [2 * a * c + 2 * b * d, 2 * b * c - 2 * a * d, a**2 + b**2 - c**2 - d**2],
        ],
        dtype=float,
    )
    return rotation


def _convert_so3_to_su2(matrix: np.ndarray) -> np.ndarray:
    """Converts an SO(3)-matrix to a corresponding SU(2)-matrix.

    Args:
        matrix: SO(3)-matrix to convert.

    Returns:
        SU(2)-matrix corresponding to SO(3)-matrix ``matrix``.

    Raises:
        ValueError: if ``matrix`` is not an SO(3)-matrix.
    """
    _check_is_so3(matrix)
    return _compute_su2_from_euler_angles(_compute_euler_angles_from_so3(matrix))


def _check_is_su2(matrix: np.ndarray) -> None:
    """Check whether ``matrix`` is SU(2), otherwise raise an error."""
    if matrix.shape != (2, 2):
        raise ValueError(f"Matrix must have shape (2, 2) but has {matrix.shape}.")

    if abs(np.linalg.det(matrix) - 1) > 1e-4:
        raise ValueError(f"Determinant of matrix must be 1, but is {np.linalg.det(matrix)}.")


def _check_is_so3(matrix: np.ndarray) -> None:
    """Check whether ``matrix`` is SO(3), otherwise raise an error."""
    if matrix.shape != (3, 3):
        raise ValueError(f"Matrix must have shape (3, 3) but has {matrix.shape}.")

    if abs(np.linalg.det(matrix) - 1) > 1e-4:
        raise ValueError(f"Determinant of matrix must be 1, but is {np.linalg.det(matrix)}.")
