code: def _tapes_data_hardware(
      tape, operation, key, num_split_times, use_broadcasting
  ):
      """Create tapes and gradient data for a trainable parameter of a HardwareHamiltonian,
      taking into account its reordering function.
  
      Args:
          tape (QuantumScript): Tape for which to compute the stochastic pulse parameter-shift
              gradient tapes.
          operation (tuple[Operation, int, int]): Information about the pulse operation to be
              shifted. The first entry is the operation itself, the second entry is its position
              in the ``tape``, and the third entry is the index of the differentiated parameter
              within the ``HardwareHamiltonian`` of the operation.
          key (tuple[int]): Randomness key to create spliting times in ``_generate_tapes_and_cjacs``
          num_split_times (int): Number of splitting times at which to create shifted tapes for
              the stochastic shift rule.
          use_broadcasting (bool): Whether to use broadcasting in the shift rule or not.
  
      Returns:
          list[QuantumScript]: Gradient tapes for the indicated operation and Hamiltonian term.
          tuple: Gradient postprocessing data.
              See comment below.
  
      This function analyses the ``reorder_fn`` of the ``HardwareHamiltonian`` of the pulse
      that is being differentiated. Given a ``term_idx``, the index of the parameter
      in the Hamiltonian, stochastic parameter shift tapes are created for all terms in the
      Hamiltonian into which the parameter feeds. While this is a one-to-one relation for
      standard ``ParametrizedHamiltonian`` objects, the reordering function of
      the ``HardwareHamiltonian`` requires to create tapes for multiple Hamiltonian terms,
      and for each term ``_generate_tapes_and_cjacs`` is called.
  
      The returned gradient data has four entries:
  
        1. ``int``: Total number of tapes created for all the terms that depend on the indicated
           parameter.
        2. ``tuple[tensor_like]``: Classical Jacobians for all terms and splitting times
        3. ``float``: Prefactor for the Monte Carlo estimate of the integral in the stochastic
           shift rule.
        4. ``tuple[tensor_like]``: Parameter-shift coefficients for all terms.
  
      The tuple axes in the second and fourth entry correspond to the different terms in the
      Hamiltonian.
      """
      op, op_idx, term_idx = operation
      # Map a simple enumeration of numbers from HardwareHamiltonian input parameters to
      # ParametrizedHamiltonian parameters. This is typically a fan-out function.
      fake_params, allowed_outputs = np.arange(op.num_params), set(
          range(op.num_params)
      )
      reordered = op.H.reorder_fn(fake_params, op.H.coeffs_parametrized)
  
      def _raise():
          raise ValueError(
              "Only permutations, fan-out or fan-in functions are allowed as reordering functions "
              "in HardwareHamiltonians treated by stoch_pulse_grad. The reordering function of "
              f"{op.H} mapped {fake_params} to {reordered}."
          )
  
      cjacs, tapes, psr_coeffs = [], [], []
      for coeff_idx, x in enumerate(reordered):
          # Find out whether the value term_idx, corresponding to the current parameter of interest,
          # has been mapped to x (for scalar x) or into x (for 1d x). If so, generate tapes and data
          # Also check that only allowed outputs have been produced by the reordering function.
          if not hasattr(x, "__len__"):
              if x not in allowed_outputs:
                  _raise()
              if x != term_idx:
                  continue
              cjac_idx = None
          else:
              if not all(_x in list(range(op.num_params)) for _x in x):
                  _raise()
              if term_idx not in x:
                  continue
              cjac_idx = np.argwhere([_x == term_idx for _x in x])[0][0]
  
          _operation = (op, op_idx, coeff_idx)
          # Overwriting int_prefactor does not matter, it is equal for all parameters in this op,
          # because it only consists of the duration `op.t[-1]-op.t[0]` and `num_split_times`
          (
              _tapes,
              _cjacs,
              int_prefactor,
              _psr_coeffs,
          ) = _generate_tapes_and_cjacs(
              tape,
              _operation,
              key,
              num_split_times,
              use_broadcasting,
              cjac_idx,
          )
          cjacs.append(qml.math.stack(_cjacs))
          tapes.extend(_tapes)
          psr_coeffs.append(_psr_coeffs)
  
      # The fact that psr_coeffs are a tuple only for hardware Hamiltonian generators will be
      # used in `_parshift_and_integrate`.
      data = (
          len(tapes),
          tuple(cjacs),
          int_prefactor,
          tuple(psr_coeffs),
      )
      return tapes, data
  
file_path: pennylane/pennylane/gradients/pulse_gradient.py
imports: from typing import Sequence, Callable
  from functools import partial
  import warnings
  import numpy
  import pennylane
  from pennylane.pulse import ParametrizedEvolution, HardwareHamiltonian
  from pennylane import transform
  from .parameter_shift import _make_zero_rep
  from .general_shift_rules import eigvals_to_frequencies, generate_shift_rule
  from .gradient_transform import _all_zero_grad, assert_no_state_returns, assert_no_tape_batching, assert_no_variance, choose_trainable_params, find_and_validate_gradient_methods, _no_trainable_grad, reorder_grads
label_desc: Use image node_id 8 for calling a global function with example usage: _tapes_data_hardware(tape, operation, key, num_split_times, use_broadcasting) and returns: tapes,  data