code: def _grad_method_cv(tape, idx):
      """Determine the best CV parameter-shift gradient recipe for a given
      parameter index of a tape.
  
      Args:
          tape (.QuantumTape): input tape
          idx (int): positive integer corresponding to the parameter location
              on the tape to inspect
  
      Returns:
          str: a string containing either ``"A"`` (for first-order analytic method),
              ``"A2"`` (second-order analytic method), ``"F"`` (finite differences),
              or ``"0"`` (constant parameter).
      """
  
      par_info = tape._par_info[idx]
      op = par_info["op"]
  
      if op.grad_method in (None, "F"):
          return op.grad_method
  
      if op.grad_method != "A":
          raise ValueError(
              f"Operation {op} has unknown gradient method {op.grad_method}"
          )
  
      # Operation supports the CV parameter-shift rule.
      # Create an empty list to store the 'best' partial derivative method
      # for each observable
      best = []
  
      for m in tape.measurements:
          if isinstance(m, ProbabilityMP) or (
              m.obs.ev_order not in (1, 2)
          ):
              # Higher-order observables (including probability) only support finite differences.
              best.append("F")
              continue
  
          # get the set of operations betweens the operation and the observable
          op_or_mp = tape[par_info["op_idx"]]
          ops_between = tape.graph.nodes_between(op_or_mp, m)
  
          if not ops_between:
              # if there is no path between the operation and the observable,
              # the operator has a zero gradient.
              best.append("0")
              continue
  
          # For parameter-shift compatible CV gates, we need to check both the
          # intervening gates, and the type of the observable.
          best_method = "A"
  
          ops_between = [
              o.obs if isinstance(o, MeasurementProcess) else o
              for o in ops_between
          ]
          if any(not k.supports_heisenberg for k in ops_between):
              # non-Gaussian operators present in-between the operation
              # and the observable. Must fallback to numeric differentiation.
              best_method = "F"
  
          elif m.obs.ev_order == 2:
              if isinstance(m, ExpectationMP):
                  # If the observable is second-order, we must use the second-order
                  # CV parameter shift rule
                  best_method = "A2"
  
              elif isinstance(m, VarianceMP):
                  # we only support analytic variance gradients for
                  # first-order observables
                  best_method = "F"
  
          best.append(best_method)
  
      if all(k == "0" for k in best):
          # if the operation is independent of *all* observables
          # in the circuit, the gradient will be 0
          return "0"
  
      if "F" in best:
          # one non-analytic observable path makes the whole operation
          # gradient method fallback to finite-difference
          return "F"
  
      if "A2" in best:
          # one second-order observable makes the whole operation gradient
          # require the second-order parameter-shift rule
          return "A2"
  
      return "A"
  
imports: from typing import Sequence, Callable
  import itertools
  from functools import partial
  import warnings
  import numpy
  import pennylane
  from pennylane.measurements import ExpectationMP, ProbabilityMP, StateMP, VarianceMP, MeasurementProcess
  from pennylane import transform
  from pennylane.transforms.tape_expand import expand_invalid_trainable
  from pennylane.gradients.gradient_transform import _contract_qjac_with_cjac, choose_trainable_params, _validate_gradient_methods
  from .finite_difference import finite_diff
  from .general_shift_rules import generate_shifted_tapes, process_shifts
  from .gradient_transform import _no_trainable_grad
  from .parameter_shift import _get_operation_recipe, expval_param_shift
label_desc: Use image node_id 1 for calling a global function with example usage: _grad_method_cv(tape, idx) and returns: str, op, str, str, str