class_docstr: A symbolic operator representing the exponential of a operator.
  
  Args:
      base (~.operation.Operator): The Operator to be exponentiated
      coeff=1 (Number): A scalar coefficient of the operator.
      num_steps (int): The number of steps used in the decomposition of the exponential operator,
          also known as the Trotter number. If this value is `None` and the Suzuki-Trotter
          decomposition is needed, an error will be raised.
      id (str): id for the Exp operator. Default is None.
  
  **Example**
  
  This symbolic operator can be used to make general rotation operators:
  
  >>> x = np.array(1.23)
  >>> op = Exp( qml.PauliX(0), -0.5j * x)
  >>> qml.math.allclose(op.matrix(), qml.RX(x, wires=0).matrix())
  True
  
  This can even be used for more complicated generators:
  
  >>> t = qml.PauliX(0) @ qml.PauliX(1) + qml.PauliY(0) @ qml.PauliY(1)
  >>> isingxy = Exp(t, 0.25j * x)
  >>> qml.math.allclose(isingxy.matrix(), qml.IsingXY(x, wires=(0,1)).matrix())
  True
  
  If the coefficient is purely imaginary and the base operator is Hermitian, then
  the gate can be used in a circuit, though it may not be supported by the device and
  may not be differentiable.
  
  >>> @qml.qnode(qml.device('default.qubit', wires=1))
  ... def circuit(x):
  ...     Exp(qml.PauliX(0), -0.5j * x)
  ...     return qml.expval(qml.PauliZ(0))
  >>> print(qml.draw(circuit)(1.23))
  0: ──Exp─┤  <Z>
  
  If the base operator is Hermitian and the coefficient is real, then the ``Exp`` operator
  can be measured as an observable:
  
  >>> obs = Exp(qml.PauliZ(0), 3)
  >>> @qml.qnode(qml.device('default.qubit', wires=1))
  ... def circuit():
  ...     return qml.expval(obs)
  >>> circuit()
  tensor(20.08553692, requires_grad=True)
docstr: Decompose the exponential of ``base`` multiplied by ``coeff``.
  
  Args:
      base (Operator): exponentiated operator
      coeff (complex): coefficient multiplying the exponentiated operator
  
  Returns:
      List[Operator]: decomposition
docstr_tok: ["Decompose","the","exponential","of","``","base","``","multiplied","by","``","coeff","``",".","Args",":","base","(","Operator",")",":","exponentiated","operator","coeff","(","complex",")",":","coefficient","multiplying","the","exponentiated","operator","Returns",":","List","[","Operator","]",":","decomposition"]
code: def _recursive_decomposition(self, base: Operator, coeff: complex):
      """Decompose the exponential of ``base`` multiplied by ``coeff``.
  
      Args:
          base (Operator): exponentiated operator
          coeff (complex): coefficient multiplying the exponentiated operator
  
      Returns:
          List[Operator]: decomposition
      """
      if isinstance(base, Tensor) and len(base.wires) != len(base.obs):
          raise DecompositionUndefinedError(
              "Unable to determine if the exponential has a decomposition "
              "when the base operator is a Tensor object with overlapping wires. "
              f"Received base {base}."
          )
  
      # Change base to `Sum`/`Prod`
      if isinstance(base, Hamiltonian):
          base = qml.dot(base.coeffs, base.ops)
      elif isinstance(base, Tensor):
          base = qml.prod(*base.obs)
  
      if isinstance(base, SProd):
          return self._recursive_decomposition(
              base.base, base.scalar * coeff
          )
  
      if self.num_steps is not None and isinstance(
          base, (Hamiltonian, Sum)
      ):
          # Apply trotter decomposition
          coeffs = (
              base.coeffs
              if isinstance(base, Hamiltonian)
              else [1] * len(base)
          )
          coeffs = [c * coeff for c in coeffs]
          ops = (
              base.ops
              if isinstance(base, Hamiltonian)
              else base.operands
          )
          return self._trotter_decomposition(ops, coeffs)
  
      # Store operator classes with generators
      has_generator_types = []
      has_generator_types_anywires = []
      for op_name in qml.ops.qubit.__all__:  # pylint:disable=no-member
          op_class = getattr(
              qml.ops.qubit, op_name
          )  # pylint:disable=no-member
          if op_class.has_generator:
              if op_class.num_wires == AnyWires:
                  has_generator_types_anywires.append(op_class)
              elif op_class.num_wires == len(base.wires):
                  has_generator_types.append(op_class)
      # Ensure op_class.num_wires == base.num_wires before op_class.num_wires == AnyWires
      has_generator_types.extend(has_generator_types_anywires)
  
      for op_class in has_generator_types:
          # PauliRot and PCPhase have different positional args
          if op_class not in {qml.PauliRot, qml.PCPhase}:
              g, c = qml.generator(op_class)(coeff, base.wires)
              # Some generators are not wire-ordered (e.g. OrbitalRotation)
              new_g = qml.map_wires(g, dict(zip(g.wires, base.wires)))
  
              if qml.equal(base, new_g) and math.real(coeff) == 0:
                  coeff = math.real(
                      -1j / c * coeff
                  )  # cancel the coefficients added by the generator
                  return [op_class(coeff, g.wires)]
  
      if qml.pauli.is_pauli_word(base) and math.real(coeff) == 0:
          # Check if the exponential can be decomposed into a PauliRot gate
          return self._pauli_rot_decomposition(base, coeff)
  
      error_msg = (
          f"The decomposition of the {self} operator is not defined. "
      )
  
      if not self.num_steps:  # if num_steps was not set
          error_msg += (
              "Please set a value to ``num_steps`` when instantiating the ``Exp`` operator "
              "if a Suzuki-Trotter decomposition is required. "
          )
  
      if math.real(self.coeff) != 0 and self.base.is_hermitian:
          error_msg += "Decomposition is not defined for real coefficients of hermitian operators."
  
      raise DecompositionUndefinedError(error_msg)
  
imports: from typing import List
  from warnings import warn
  import numpy
  from scipy.sparse.linalg import expm
  import pennylane
  from pennylane import math
  from pennylane.math import expand_matrix
  from pennylane.operation import AnyWires, DecompositionUndefinedError, GeneratorUndefinedError, Operation, Operator, OperatorPropertyUndefined, Tensor
  from pennylane.ops.qubit import Hamiltonian
  from pennylane.wires import Wires
  from .sprod import SProd
  from .sum import Sum
  from .symbolicop import ScalarSymbolicOp
label_desc: Use image node_id 12 for calling the Exp obj's underlying member method code with example usage: obj._recursive_decomposition(base, coeff) and returns: self, self, self, list