class_docstr: Test the torch interface execute function
  integrates well for both forward and backward execution
class_docstr_tok: ["Test","the","torch","interface","execute","function","integrates","well","for","both","forward","and","backward","execution"]
docstr: Test that operation and nested tape expansion
  is differentiable
code: def test_differentiable_expand(
      self, torch_device, execute_kwargs, tol
  ):
      """Test that operation and nested tape expansion
      is differentiable"""
  
      class U3(qml.U3):
          def decomposition(self):
              theta, phi, lam = self.data
              wires = self.wires
              return [
                  qml.Rot(lam, theta, -lam, wires=wires),
                  qml.PhaseShift(phi + lam, wires=wires),
              ]
  
      dev = qml.device("default.qubit.legacy", wires=1)
      a = np.array(0.1)
      p_val = [0.1, 0.2, 0.3]
      p = torch.tensor(p_val, requires_grad=True, device=torch_device)
  
      with qml.queuing.AnnotatedQueue() as q:
          qml.RX(a, wires=0)
          U3(p[0], p[1], p[2], wires=0)
          qml.expval(qml.PauliX(0))
  
      tape = qml.tape.QuantumScript.from_queue(q)
  
      res = execute([tape], dev, **execute_kwargs)[0]
  
      expected = torch.tensor(
          np.cos(a) * np.cos(p_val[1]) * np.sin(p_val[0])
          + np.sin(a)
          * (
              np.cos(p_val[2]) * np.sin(p_val[1])
              + np.cos(p_val[0]) * np.cos(p_val[1]) * np.sin(p_val[2])
          ),
          dtype=res.dtype,
          device=torch_device,
      )
      assert torch.allclose(res.detach(), expected, atol=tol, rtol=0)
  
      res.backward()
      expected = torch.tensor(
          [
              np.cos(p_val[1])
              * (
                  np.cos(a) * np.cos(p_val[0])
                  - np.sin(a) * np.sin(p_val[0]) * np.sin(p_val[2])
              ),
              np.cos(p_val[1]) * np.cos(p_val[2]) * np.sin(a)
              - np.sin(p_val[1])
              * (
                  np.cos(a) * np.sin(p_val[0])
                  + np.cos(p_val[0]) * np.sin(a) * np.sin(p_val[2])
              ),
              np.sin(a)
              * (
                  np.cos(p_val[0]) * np.cos(p_val[1]) * np.cos(p_val[2])
                  - np.sin(p_val[1]) * np.sin(p_val[2])
              ),
          ],
          dtype=p.grad.dtype,
          device=torch_device,
      )
      assert torch.allclose(p.grad, expected, atol=tol, rtol=0)
  
imports: import numpy
  import pytest
  import pennylane
  from pennylane.gradients import finite_diff, param_shift
  from pennylane import execute
label_desc: Use image node_id 9 for calling the TestTorchExecuteIntegration obj's underlying member method code with example usage: obj.test_differentiable_expand(torch_device, execute_kwargs, tol) and returns: list