code: def generate_shift_rule(frequencies, shifts=None, order=1):
      r"""Computes the parameter shift rule for a unitary based on its generator's eigenvalue
      frequency spectrum.
  
      To compute gradients of circuit parameters in variational quantum algorithms, expressions for
      cost function first derivatives with respect to the variational parameters can be cast into
      linear combinations of expectation values at shifted parameter values. The coefficients and
      shifts defining the linear combination can be obtained from the unitary generator's eigenvalue
      frequency spectrum. Details can be found in
      `Wierichs et al. (2022) <https://doi.org/10.22331/q-2022-03-30-677>`__.
  
      Args:
          frequencies (tuple[int or float]): The tuple of eigenvalue frequencies. Eigenvalue
              frequencies are defined as the unique positive differences obtained from a set of
              eigenvalues.
          shifts (tuple[int or float]): the tuple of shift values. If unspecified,
              equidistant shifts are assumed. If supplied, the length of this tuple should match the
              number of given frequencies.
          order (int): the order of differentiation to compute the shift rule for
  
      Returns:
          tuple: a tuple of coefficients and shifts describing the gradient rule for the
          parameter-shift method. For parameter :math:`\phi`, the coefficients :math:`c_i` and the
          shifts :math:`s_i` combine to give a gradient rule of the following form:
  
          .. math:: \frac{\partial}{\partial\phi}f = \sum_{i} c_i f(\phi + s_i).
  
          where :math:`f(\phi) = \langle 0|U(\phi)^\dagger \hat{O} U(\phi)|0\rangle`
          for some observable :math:`\hat{O}` and the unitary :math:`U(\phi)=e^{iH\phi}`.
  
      Raises:
          ValueError: if ``frequencies`` is not a list of unique positive values, or if ``shifts``
              (if specified) is not a list of unique values the same length as ``frequencies``.
  
      **Examples**
  
      An example of obtaining the frequencies from generator eigenvalues, and obtaining the parameter
      shift rule:
  
      >>> eigvals = (-0.5, 0, 0, 0.5)
      >>> frequencies = eigvals_to_frequencies(eigvals)
      >>> generate_shift_rule(frequencies)
      array([[ 0.4267767 ,  1.57079633],
             [-0.4267767 , -1.57079633],
             [-0.0732233 ,  4.71238898],
             [ 0.0732233 , -4.71238898]])
  
      An example with explicitly specified shift values:
  
      >>> frequencies = (1, 2, 4)
      >>> shifts = (np.pi / 3, 2 * np.pi / 3, np.pi / 4)
      >>> generate_shift_rule(frequencies, shifts)
      array([[ 3.        ,  0.78539816],
             [-3.        , -0.78539816],
             [-2.09077028,  1.04719755],
             [ 2.09077028, -1.04719755],
             [ 0.2186308 ,  2.0943951 ],
             [-0.2186308 , -2.0943951 ]])
  
      Higher order shift rules (corresponding to the :math:`n`-th derivative of the parameter) can be
      requested via the ``order`` argument. For example, to extract the second order shift rule for a
      gate with generator :math:`X/2`:
  
      >>> eigvals = (0.5, -0.5)
      >>> frequencies = eigvals_to_frequencies(eigvals)
      >>> generate_shift_rule(frequencies, order=2)
      array([[-0.5       ,  0.        ],
             [ 0.5       , -3.14159265]])
  
      This corresponds to the shift rule
      :math:`\frac{\partial^2 f}{\partial phi^2} = \frac{1}{2} \left[f(\phi) - f(\phi-\pi)\right]`.
      """
      frequencies = tuple(f for f in frequencies if f > 0)
      rule = _get_shift_rule(frequencies, shifts=shifts)
  
      if order > 1:
          T = frequencies_to_period(frequencies)
          rule = _iterate_shift_rule(rule, order, period=T)
  
      return process_shifts(rule, tol=1e-10)
  
imports: import functools
  import itertools
  import numbers
  import warnings
  import numpy
  import pennylane
  from pennylane.measurements import MeasurementProcess
  from pennylane.ops.functions import bind_new_parameters
  from pennylane.tape import QuantumScript
label_desc: Use image node_id 8 for calling a global function with example usage: generate_shift_rule(frequencies, shifts, order) and returns: process_shifts