code: def simplify_pauli_sum(pauli_sum):
      
  
      # You might want to use a defaultdict(list) here, but don't because
      # we want to do our best to preserve the order of terms.
      like_terms = OrderedDict()
      for term in pauli_sum.terms:
          key = term.operations_as_set()
          if key in like_terms:
              like_terms[key].append(term)
          else:
              like_terms[key] = [term]
  
      terms = []
      for term_list in like_terms.values():
          first_term = term_list[0]
          if len(term_list) == 1 and not np.isclose(first_term.coefficient, 0.0):
              terms.append(first_term)
          else:
              coeff = sum(t.coefficient for t in term_list)
              for t in term_list:
                  if list(t._ops.items()) != list(first_term._ops.items()):
                      warnings.warn("The term {} will be combined with {}, but they have different "
                                    "orders of operations. This doesn't matter for QVM or "
                                    "wavefunction simulation but may be important when "
                                    "running on an actual device."
                                    .format(t.id(sort_ops=False), first_term.id(sort_ops=False)))
  
              if not np.isclose(coeff, 0.0):
                  terms.append(term_with_coeff(term_list[0], coeff))
      return PauliSum(terms)
docs: Simplify the sum of Pauli operators according to Pauli algebra rules.