class_docstr: Rotoselect gradient-free optimizer.
  
  The Rotoselect optimizer minimizes an objective function with respect to the rotation gates and
  parameters of a quantum circuit without the need for calculating the gradient of the function.
  The algorithm works by updating the parameters :math:`\theta = \theta_1, \dots, \theta_D`
  and rotation gate choices :math:`R = R_1,\dots,R_D` one at a time according to a closed-form
  expression for the optimal value of the :math:`d^{th}` parameter :math:`\theta^*_d` when the
  other parameters and gate choices are fixed:
  
  .. math:: \theta^*_d = \underset{\theta_d}{\text{argmin}}\left<H\right>_{\theta_d}
            = -\frac{\pi}{2} - \text{arctan2}\left(2\left<H\right>_{\theta_d=0}
            - \left<H\right>_{\theta_d=\pi/2} - \left<H\right>_{\theta_d=-\pi/2},
            \left<H\right>_{\theta_d=\pi/2} - \left<H\right>_{\theta_d=-\pi/2}\right),
  
  where :math:`\left<H\right>_{\theta_d}` is the expectation value of the objective function
  optimized over the parameter :math:`\theta_d`. :math:`\text{arctan2}(x, y)` computes the
  element-wise arc tangent of :math:`x/y` choosing the quadrant correctly, avoiding, in
  particular, division-by-zero when :math:`y = 0`.
  
  Which parameters and gates that should be optimized over is decided in the user-defined cost
  function, where :math:`R` is a list of parametrized rotation gates in a quantum circuit, along
  with their respective parameters :math:`\theta` for the circuit and its gates. Note that the
  number of generators should match the number of parameters.
  
  The algorithm is described in further detail in
  `Ostaszewski et al. (2021) <https://doi.org/10.22331/q-2021-01-28-391>`_.
  
  Args:
      possible_generators (list[~.Operation]): List containing the possible
          ``pennylane.ops.qubit`` operators that are allowed in the circuit.
          Default is the set of Pauli rotations :math:`\{R_x, R_y, R_z\}`.
  
  **Example:**
  
  Initialize the Rotoselect optimizer, set the initial values of  the weights ``x``,
  choose the initial generators, and set the number of steps to optimize over.
  
  >>> opt = qml.optimize.RotoselectOptimizer()
  >>> x = [0.3, 0.7]
  >>> generators = [qml.RX, qml.RY]
  >>> n_steps = 10
  
  Set up the PennyLane circuit using the ``default.qubit`` simulator device.
  
  >>> dev = qml.device("default.qubit", shots=None, wires=2)
  >>> @qml.qnode(dev)
  ... def circuit(params, generators=None):  # generators will be passed as a keyword arg
  ...     generators[0](params[0], wires=0)
  ...     generators[1](params[1], wires=1)
  ...     qml.CNOT(wires=[0, 1])
  ...     return qml.expval(qml.PauliZ(0)), qml.expval(qml.PauliX(1))
  
  Define a cost function based on the above circuit.
  
  >>> def cost(x, generators):
  ...     Z_1, X_2 = circuit(x, generators=generators)
  ...     return 0.2 * Z_1 + 0.5 * X_2
  
  Run the optimization step-by-step for ``n_steps`` steps.
  
  >>> cost_rotosel = []
  >>> for _ in range(n_steps):
  ...     cost_rotosel.append(cost(x, generators))
  ...     x, generators = opt.step(cost, x, generators)
  
  The optimized values for x should now be stored in ``x`` together with the optimal gates for
  the circuit, while steps-vs-cost can be seen by plotting ``cost_rotosel``.
docstr: Update trainable arguments with one step of the optimizer and return the corresponding
  objective function value prior to the step.
  
  Args:
      objective_fn (function): The objective function for optimization. It must have the
          signature ``objective_fn(x, generators=None)`` with a sequence of the values ``x``
          and a list of the gates ``generators`` as inputs, returning a single value.
      x (Union[Sequence[float], float]): sequence containing the initial values of the
          variables to be optimized over or a single float with the initial value
      generators (list[~.Operation]): list containing the initial ``pennylane.ops.qubit``
          operators to be used in the circuit and optimized over
      **kwargs : variable length of keyword arguments for the objective function.
  
  Returns:
      tuple: the new variable values :math:`x^{(t+1)}`, the new generators, and the objective
      function output prior to the step
code: def step_and_cost(self, objective_fn, x, generators, **kwargs):
      """Update trainable arguments with one step of the optimizer and return the corresponding
      objective function value prior to the step.
  
      Args:
          objective_fn (function): The objective function for optimization. It must have the
              signature ``objective_fn(x, generators=None)`` with a sequence of the values ``x``
              and a list of the gates ``generators`` as inputs, returning a single value.
          x (Union[Sequence[float], float]): sequence containing the initial values of the
              variables to be optimized over or a single float with the initial value
          generators (list[~.Operation]): list containing the initial ``pennylane.ops.qubit``
              operators to be used in the circuit and optimized over
          **kwargs : variable length of keyword arguments for the objective function.
  
      Returns:
          tuple: the new variable values :math:`x^{(t+1)}`, the new generators, and the objective
          function output prior to the step
      """
      x_new, generators = self.step(
          objective_fn, x, generators, **kwargs
      )
  
      return x_new, generators, objective_fn(x, generators, **kwargs)
  
file_path: pennylane/pennylane/optimize/rotoselect.py
imports: import numpy
  import pennylane
  from pennylane.utils import _flatten, unflatten
label_desc: Use image node_id 2 for calling the RotoselectOptimizer obj's underlying member method code with example usage: obj.step_and_cost(objective_fn, x, generators) and returns: x_new,  generators,  objective_fn