code: def _pauli_mult(p1, p2):
      r"""Return the result of multiplication between two tensor products of Pauli operators.
  
      The Pauli operator :math:`(P_0)` is denoted by [(0, 'P')], where :math:`P` represents
      :math:`X`, :math:`Y` or :math:`Z`.
  
      Args:
          p1 (list[tuple[int, str]]): the first tensor product of Pauli operators
          p2 (list[tuple[int, str]]): the second tensor product of Pauli operators
  
      Returns
          tuple(list[tuple[int, str]], complex): list of the Pauli operators and the coefficient
  
      **Example**
  
      >>> p1 = [(0, "X"), (1, "Y")]  # X_0 @ Y_1
      >>> p2 = [(0, "X"), (2, "Y")]  # X_0 @ Y_2
      >>> _pauli_mult(p1, p2)
      ([(2, "Y"), (1, "Y")], 1.0) # p1 @ p2 = X_0 @ Y_1 @ X_0 @ Y_2
      """
  
      warn(
          "_pauli_mult is deprecated. Instead, please use the "
          "PauliWord class, or regular PennyLane operators.",
          qml.PennyLaneDeprecationWarning,
      )
      c = 1.0
  
      t1 = [t[0] for t in p1]
      t2 = [t[0] for t in p2]
  
      k = []
  
      for i in p1:
          if i[0] in t1 and i[0] not in t2:
              k.append((i[0], pauli_mult_dict[i[1]]))
          for j in p2:
              if j[0] in t2 and j[0] not in t1:
                  k.append((j[0], pauli_mult_dict[j[1]]))
  
              if i[0] == j[0]:
                  k.append((i[0], pauli_mult_dict[i[1] + j[1]]))
                  if i[1] + j[1] in pauli_coeff:
                      c = c * pauli_coeff[i[1] + j[1]]
  
      for item in k:
          k_ = [i for i, x in enumerate(k) if x == item]
          if len(k_) >= 2:
              for j in k_[::-1][:-1]:
                  del k[j]
  
      return k, c
  
imports: from functools import lru_cache, reduce, singledispatch
  from itertools import product
  from typing import List, Union
  from warnings import warn
  import numpy
  import pennylane
  from pennylane.operation import Tensor
  from pennylane.ops import Hamiltonian, Identity, PauliX, PauliY, PauliZ, Prod, SProd
  from pennylane.tape import OperationRecorder
  from pennylane.wires import Wires
label_desc: Use image node_id 29 for calling a global function with example usage: _pauli_mult(p1, p2) and returns: k,  c