code: def _generate_tapes_and_coeffs(tape, idx, atol, cache):
      """Compute the modified tapes and coefficients required to compute the pulse generator
      derivative of a tape with respect to an indicated trainable parameter.
  
      Args:
          tape (`~.QuantumTape`): Tape to differentiate.
          idx (int): Index (referring to ``tape.trainable_parameters``) of the parameter
              with respect to which to differentiate.
          atol (float): absolute tolerance used to determine whether a coefficient is zero.
          cache (dict): Caching dictionary that allows to skip adding duplicate modified tapes.
  
      Returns:
          list[`~.QuantumScript`]: Modified tapes to be added to the pulse generator differentiation
              tapes. It is an empty list if modified tapes were already created for another
              parameter of the pulse of interest.
          tuple[int, int, tensor_like]: Gradient computation data, consisting of the start and end
              indices into the total list of tapes as well as the coefficients that need to be
              contracted with the corresponding results to obtain the partial derivative with respect
              to the indicated trainable parameter.
          dict: Input ``cache`` dictionary. If the cache lookup missed, the cache is extended by one
              entry and its entry ``"total_num_tapes"`` is increased by the number of created tapes.
      """
      # ``op`` is the operation into which the trainable tape parameter with index ``idx`` feeds.
      # ``op_idx`` is the index of ``op`` within all tape operations. ``term_idx`` is the index
      # of the trainable tape parameter within ``op``.
      op, op_idx, term_idx = tape.get_operation(idx)
      if op_idx in cache:
          # operation was analyzed and tapes were added before. Get tape indices and coefficients.
          start, end, all_coeffs = cache[op_idx]
          return [], (start, end, all_coeffs[term_idx]), cache
  
      if not isinstance(op, ParametrizedEvolution):
          # only ParametrizedEvolution can be treated with this gradient transform
          raise ValueError(
              "pulse_odegen does not support differentiating parameters of "
              f"other operations than pulses, but received operation {op}."
          )
  
      num_wires = len(op.wires)
      # Compute the effective generators of the given pulse _for all parameters_.
      generators = _one_parameter_generators(op)
      # Obtain one-parameter-group coefficients in Pauli basis and filter for non-zero coeffs
      all_coeffs = _one_parameter_paulirot_coeffs(generators, num_wires)
      all_coeffs, pauli_words = _nonzero_coeffs_and_words(
          all_coeffs, num_wires, atol
      )
      # create PauliRot gates for each Pauli word (with a non-zero coefficient) and for both shifts
      pauli_rots = [
          qml.PauliRot(angle, word, wires=op.wires)
          for word in pauli_words
          for angle in [np.pi / 2, -np.pi / 2]
      ]
      # create tapes with the above PauliRot gates inserted, one per tape
      tapes = _insert_op(tape, pauli_rots, op_idx)
      # get the previous total number of tapes from the cache and determine start and end indices
      end = (start := cache["total_num_tapes"]) + (
          num_tapes := len(tapes)
      )
      # store the tape indices and coefficients for this operation, to be retrieved from the cache
      # for other parameters of the same operation, see first if clause above.
      cache[op_idx] = (start, end, all_coeffs)
      # increment total number of tapes
      cache["total_num_tapes"] += num_tapes
      # Return the start and end pointer for the created tapes/the corresponding results.
      # Also return the coefficients for the current trainable parameter of the pulse
      return tapes, (start, end, all_coeffs[term_idx]), cache
  
file_path: pennylane/pennylane/gradients/pulse_gradient_odegen.py
imports: from typing import Callable, Sequence
  from functools import partial
  import numpy
  import pennylane
  from pennylane.pulse import ParametrizedEvolution
  from pennylane.ops.qubit.special_unitary import pauli_basis_strings, _pauli_decompose
  from pennylane import transform
  from .parameter_shift import _make_zero_rep
  from .pulse_gradient import _assert_has_jax, raise_pulse_diff_on_qnode
  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 5 for calling a global function with example usage: _generate_tapes_and_coeffs(tape, idx, atol, cache) and returns: tapes,  start,  end,  all_coeffs,  cache, list,  start,  end,  all_coeffs,  cache