code: def _generalized_pauli_decompose(
      matrix,
      hide_identity=False,
      wire_order=None,
      pauli=False,
      padding=False,
  ) -> Tuple[qml.typing.TensorLike, list]:
      r"""Decomposes any matrix into a linear combination of Pauli operators.
  
      This method converts any matrix to a weighted sum of Pauli words acting on :math:`n` qubits
      in time :math:`O(n 4^n)`. The input matrix is first padded with zeros if its dimensions are not
      :math:`2^n\times 2^n` and written as a quantum state in the computational basis following the
      `channel-state duality <https://en.wikipedia.org/wiki/Channel-state_duality>`_.
      A Bell basis transformation is then performed using the
      `Walsh-Hadamard transform <https://en.wikipedia.org/wiki/Hadamard_transform>`_, after which
      coefficients for each of the :math:`4^n` Pauli words are computed while accounting for the
      phase from each ``PauliY`` term occurring in the word.
  
      Args:
          matrix (tensor_like[complex]): any matrix M, the keyword argument ``padding=True``
              should be provided if the dimension of M is not :math:`2^n\times 2^n`.
          hide_identity (bool): does not include the Identity observable within
              the tensor products of the decomposition if ``True``.
          wire_order (list[Union[int, str]]): the ordered list of wires with respect
              to which the operator is represented as a matrix.
          pauli (bool): return a PauliSentence instance if ``True``.
          padding (bool): makes the function compatible with rectangular matrices and square matrices
              that are not of shape :math:`2^n\times 2^n` by padding them with zeros if ``True``.
  
      Returns:
          Tuple[qml.math.array[complex], list]: the matrix decomposed as a linear combination of Pauli operators
          as a tuple consisting of an array of complex coefficients and a list of corresponding Pauli terms.
  
      **Example:**
  
      We can use this function to compute the Pauli operator decomposition of an arbitrary matrix:
  
      >>> A = np.array(
      ... [[-2, -2+1j, -2, -2], [-2-1j,  0,  0, -1], [-2,  0, -2, -1], [-2, -1, -1,  1j]])
      >>> coeffs, obs = qml.pauli.conversion._generalized_pauli_decompose(A)
      >>> coeffs
      array([-1. +0.25j, -1.5+0.j  , -0.5+0.j  , -1. -0.25j, -1.5+0.j  ,
         -1. +0.j  , -0.5+0.j  ,  1. -0.j  ,  0. -0.25j, -0.5+0.j  ,
         -0.5+0.j  ,  0. +0.25j])
      >>> obs
      [Identity(wires=[0]) @ Identity(wires=[1]),
      Identity(wires=[0]) @ PauliX(wires=[1]),
      Identity(wires=[0]) @ PauliY(wires=[1]),
      Identity(wires=[0]) @ PauliZ(wires=[1]),
      PauliX(wires=[0]) @ Identity(wires=[1]),
      PauliX(wires=[0]) @ PauliX(wires=[1]),
      PauliX(wires=[0]) @ PauliZ(wires=[1]),
      PauliY(wires=[0]) @ PauliY(wires=[1]),
      PauliZ(wires=[0]) @ Identity(wires=[1]),
      PauliZ(wires=[0]) @ PauliX(wires=[1]),
      PauliZ(wires=[0]) @ PauliY(wires=[1]),
      PauliZ(wires=[0]) @ PauliZ(wires=[1])]
  
      We can also set custom wires using the ``wire_order`` argument:
  
      >>> coeffs, obs = qml.pauli.conversion._generalized_pauli_decompose(A, wire_order=['a', 'b'])
      >>> obs
      [Identity(wires=['a']) @ Identity(wires=['b']),
      Identity(wires=['a']) @ PauliX(wires=['b']),
      Identity(wires=['a']) @ PauliY(wires=['b']),
      Identity(wires=['a']) @ PauliZ(wires=['b']),
      PauliX(wires=['a']) @ Identity(wires=['b']),
      PauliX(wires=['a']) @ PauliX(wires=['b']),
      PauliX(wires=['a']) @ PauliZ(wires=['b']),
      PauliY(wires=['a']) @ PauliY(wires=['b']),
      PauliZ(wires=['a']) @ Identity(wires=['b']),
      PauliZ(wires=['a']) @ PauliX(wires=['b']),
      PauliZ(wires=['a']) @ PauliY(wires=['b']),
      PauliZ(wires=['a']) @ PauliZ(wires=['b'])]
  
      .. details::
          :title: Advanced Usage Details
          :href: usage-decompose-operation
  
          For non-square matrices, we need to provide the ``padding=True`` keyword argument:
  
          >>> A = np.array([[-2, -2 + 1j]])
          >>> coeffs, obs = qml.pauli.conversion._generalized_pauli_decompose(A, padding=True)
          >>> coeffs
          ([-1. +0.j , -1. +0.5j, -0.5-1.j , -1. +0.j ])
          >>> obs
          [Identity(wires=[0]), PauliX(wires=[0]), PauliY(wires=[0]), PauliZ(wires=[0])]
  
          We can also use the method within a differentiable workflow and obtain gradients:
  
          >>> A = qml.numpy.array([[-2, -2 + 1j]], requires_grad=True)
          >>> dev = qml.device("default.qubit", wires=1)
          >>> @qml.qnode(dev)
          ... def circuit(A):
          ...    coeffs, _ = qml.pauli.conversion._generalized_pauli_decompose(A, padding=True)
          ...    qml.RX(qml.math.real(coeffs[2]), 0)
          ...    return qml.expval(qml.PauliZ(0))
          >>> qml.grad(circuit)(A)
          array([[0.+0.j        , 0.+0.23971277j]])
  
      """
      # Ensuring original matrix is not manipulated and we support builtin types.
      matrix = qml.math.convert_like(
          matrix, next(iter([*matrix[0]]), [])
      )
  
      # Pad with zeros to make the matrix shape equal and a power of two.
      if padding:
          shape = qml.math.shape(matrix)
          num_qubits = int(
              qml.math.ceil(qml.math.log2(qml.math.max(shape)))
          )
          if shape[0] != shape[1] or shape[0] != 2**num_qubits:
              padd_diffs = qml.math.abs(
                  qml.math.array(shape) - 2**num_qubits
              )
              padding = (
                  ((0, padd_diffs[0]), (0, padd_diffs[1]))
                  if qml.math.get_interface(matrix) != "torch"
                  else ((padd_diffs[0], 0), (padd_diffs[1], 0))
              )
              matrix = qml.math.pad(
                  matrix, padding, mode="constant", constant_values=0
              )
  
      shape = qml.math.shape(matrix)
      if shape[0] != shape[1]:
          raise ValueError(
              f"The matrix should be square, got {shape}. Use 'padding=True' for rectangular matrices."
          )
  
      num_qubits = int(qml.math.log2(shape[0]))
      if shape[0] != 2**num_qubits:
          raise ValueError(
              f"Dimension of the matrix should be a power of 2, got {shape}. Use 'padding=True' for these matrices."
          )
  
      if wire_order is not None and len(wire_order) != num_qubits:
          raise ValueError(
              f"number of wires {len(wire_order)} is not compatible with the number of qubits {num_qubits}"
          )
  
      if wire_order is None:
          wire_order = range(num_qubits)
  
      # Permute by XORing
      indices = [qml.math.array(range(shape[0]))]
      for idx in range(shape[0] - 1):
          indices.append(
              qml.math.bitwise_xor(indices[-1], (idx + 1) ^ (idx))
          )
      term_mat = qml.math.cast(
          qml.math.stack(
              [
                  qml.math.gather(matrix[idx], indice)
                  for idx, indice in enumerate(indices)
              ]
          ),
          complex,
      )
  
      # Perform Hadamard transformation on coloumns
      hadamard_transform_mat = _walsh_hadamard_transform(
          qml.math.transpose(term_mat)
      )
  
      # Account for the phases from Y
      phase_mat = qml.math.ones(shape, dtype=complex).reshape(
          (2,) * (2 * num_qubits)
      )
      for idx in range(num_qubits):
          index = [slice(None)] * (2 * num_qubits)
          index[idx] = index[idx + num_qubits] = 1
          phase_mat[tuple(index)] *= 1j
      phase_mat = qml.math.convert_like(
          qml.math.reshape(phase_mat, shape), matrix
      )
  
      # c_00 + c_11 -> I; c_00 - c_11 -> Z; c_01 + c_10 -> X; 1j*(c_10 - c_01) -> Y
      # https://quantumcomputing.stackexchange.com/a/31790
      term_mat = qml.math.transpose(
          qml.math.multiply(hadamard_transform_mat, phase_mat)
      )
  
      # Obtain the coefficients for each Pauli word
      coeffs, obs = [], []
      for pauli_rep in product("IXYZ", repeat=num_qubits):
          bit_array = qml.math.array(
              [[(rep in "YZ"), (rep in "XY")] for rep in pauli_rep],
              dtype=int,
          ).T
          coefficient = term_mat[
              tuple(int("".join(map(str, x)), 2) for x in bit_array)
          ]
  
          if not qml.math.allclose(coefficient, 0):
              observables = (
                  [
                      (o, w)
                      for w, o in zip(wire_order, pauli_rep)
                      if o != I
                  ]
                  if hide_identity
                  and not all(t == I for t in pauli_rep)
                  else [(o, w) for w, o in zip(wire_order, pauli_rep)]
              )
              if observables:
                  coeffs.append(coefficient)
                  obs.append(observables)
  
      coeffs = qml.math.stack(coeffs)
  
      if not pauli:
          with qml.QueuingManager.stop_recording():
              obs = [
                  reduce(matmul, [op_map[o](w) for o, w in obs_term])
                  for obs_term in obs
              ]
  
      return (coeffs, obs)
  
imports: from functools import reduce, singledispatch
  from itertools import product
  from operator import matmul
  from typing import Union, Tuple
  import pennylane
  from pennylane.operation import Tensor
  from pennylane.ops import Hamiltonian, Identity, PauliX, PauliY, PauliZ, Prod, SProd, Sum
  from pennylane.ops.qubit.matrix_ops import _walsh_hadamard_transform
  from .pauli_arithmetic import I, PauliSentence, PauliWord, X, Y, Z, op_map
  from .utils import is_pauli_word
label_desc: Use image node_id 1 for calling a global function with example usage: _generalized_pauli_decompose(matrix, hide_identity, wire_order, pauli, padding) and returns: coeffs,  obs