code: def dot(
      coeffs: Sequence[Union[float, Callable]],
      ops: Sequence[Operator],
      pauli=False,
  ) -> Union[Operator, ParametrizedHamiltonian]:
      r"""Returns the dot product between the ``coeffs`` vector and the ``ops`` list of operators.
  
      This function returns the following linear combination: :math:`\sum_{k} c_k O_k`, where
      :math:`c_k` and :math:`O_k` are the elements inside the ``coeffs`` and ``ops`` arguments, respectively.
  
      Args:
          coeffs (Sequence[float, Callable]): sequence containing the coefficients of the linear combination
          ops (Sequence[Operator]): sequence containing the operators of the linear combination
          pauli (bool, optional): If ``True``, a :class:`~.PauliSentence`
              operator is used to represent the linear combination. If False, a :class:`Sum` operator
              is returned. Defaults to ``False``.
  
      Raises:
          ValueError: if the number of coefficients and operators does not match or if they are empty
  
      Returns:
          Operator or ParametrizedHamiltonian: operator describing the linear combination
  
      **Example**
  
      >>> coeffs = np.array([1.1, 2.2])
      >>> ops = [qml.PauliX(0), qml.PauliY(0)]
      >>> qml.dot(coeffs, ops)
      (1.1*(PauliX(wires=[0]))) + (2.2*(PauliY(wires=[0])))
      >>> qml.dot(coeffs, ops, pauli=True)
      1.1 * X(0)
      + 2.2 * Y(0)
  
      ``pauli=True`` can be used to construct a more efficient, simplified version of the operator.
      Note that it returns a :class:`~.PauliSentence`, which is not an :class:`~.Operator`. This
      specialized representation can be converted to an operator:
  
      >>> qml.dot([1, 2], [qml.PauliX(0), qml.PauliX(0)], pauli=True).operation()
      3.0*(PauliX(wires=[0]))
  
      Using ``pauli=True`` and then converting the result to an :class:`~.Operator` is much faster
      than using ``pauli=False``, but it only works for pauli words
      (see :func:`~.is_pauli_word`).
  
      If any of the parameters listed in ``coeffs`` are callables, the resulting dot product will be a
      :class:`~.ParametrizedHamiltonian`:
  
      >>> coeffs = [lambda p, t: p * jnp.sin(t) for _ in range(2)]
      >>> ops = [qml.PauliX(0), qml.PauliY(0)]
      >>> qml.dot(coeffs, ops)
        (<lambda>(params_0, t)*(PauliX(wires=[0])))
      + (<lambda>(params_1, t)*(PauliY(wires=[0])))
      """
  
      if len(coeffs) != len(ops):
          raise ValueError(
              "Number of coefficients and operators does not match."
          )
      if len(coeffs) == 0 and len(ops) == 0:
          raise ValueError(
              "Cannot compute the dot product of an empty sequence."
          )
  
      if any(callable(c) for c in coeffs):
          return ParametrizedHamiltonian(coeffs, ops)
  
      if pauli:
          return _pauli_dot(coeffs, ops)
  
      # When casting a Hamiltonian to a Sum, we also cast its inner Tensors to Prods
      ops = [
          qml.prod(*op.obs) if isinstance(op, Tensor) else op
          for op in ops
      ]
  
      if coeffs[0] != 1 and qml.math.allequal(coeffs[0], coeffs):
          # Coefficients have the same value (different to 1)
          return qml.s_prod(
              coeffs[0], ops[0] if len(ops) == 1 else qml.sum(*ops)
          )
  
      abs_coeffs = qml.math.abs(coeffs)
      if abs_coeffs[0] != 1 and qml.math.allequal(
          abs_coeffs[0], abs_coeffs
      ):
          # Coefficients have the same absolute value (different to 1)
          gcd = abs(coeffs[0])
          coeffs = [c / gcd for c in coeffs]
          return qml.s_prod(gcd, qml.dot(coeffs, ops))
  
      operands = [
          op if coeff == 1 else qml.s_prod(coeff, op)
          for coeff, op in zip(coeffs, ops)
      ]
      return operands[0] if len(operands) == 1 else qml.sum(*operands)
  
imports: from collections import defaultdict
  from typing import Sequence, Union, Callable
  import pennylane
  from pennylane.operation import Operator, Tensor
  from pennylane.pulse import ParametrizedHamiltonian
label_desc: Use image node_id 1 for calling a global function with example usage: dot(coeffs, ops, pauli) and returns: unknown, ParametrizedHamiltonian, _pauli_dot, qml, qml