class_docstr: Test that tape expansion within the QNode integrates correctly
  with the Torch interface
class_docstr_tok: ["Test","that","tape","expansion","within","the","QNode","integrates","correctly","with","the","Torch","interface"]
docstr: Test that if there
  are non-commuting groups and the number of shots is None
  the first and second order gradients are correctly evaluated
docstr_tok: ["Test","that","if","there","are","non-commuting","groups","and","the","number","of","shots","is","None","the","first","and","second","order","gradients","are","correctly","evaluated"]
code: def test_hamiltonian_expansion_analytic(
      self,
      dev,
      diff_method,
      grad_on_execution,
      max_diff,
      device_vjp,
      tol,
  ):
      """Test that if there
      are non-commuting groups and the number of shots is None
      the first and second order gradients are correctly evaluated"""
      kwargs = dict(
          diff_method=diff_method,
          grad_on_execution=grad_on_execution,
          max_diff=max_diff,
          interface="torch",
          device_vjp=device_vjp,
      )
      if diff_method == "adjoint":
          pytest.skip(
              "The adjoint method does not yet support Hamiltonians"
          )
      elif diff_method == "spsa":
          kwargs["sampler_rng"] = np.random.default_rng(SEED_FOR_SPSA)
          kwargs["num_directions"] = 20
          tol = TOL_FOR_SPSA
      elif diff_method == "hadamard":
          pytest.skip(
              "The hadamard method does not yet support Hamiltonians"
          )
  
      obs = [
          qml.PauliX(0),
          qml.PauliX(0) @ qml.PauliZ(1),
          qml.PauliZ(0) @ qml.PauliZ(1),
      ]
  
      @qnode(dev, **kwargs)
      def circuit(data, weights, coeffs):
          weights = torch.reshape(weights, [1, -1])
          qml.templates.AngleEmbedding(data, wires=[0, 1])
          qml.templates.BasicEntanglerLayers(weights, wires=[0, 1])
          return qml.expval(qml.Hamiltonian(coeffs, obs))
  
      d = torch.tensor(
          [0.1, 0.2], requires_grad=False, dtype=torch.float64
      )
      w = torch.tensor(
          [0.654, -0.734], requires_grad=True, dtype=torch.float64
      )
      c = torch.tensor(
          [-0.6543, 0.24, 0.54], requires_grad=True, dtype=torch.float64
      )
  
      # test output
      res = circuit(d, w, c)
  
      expected = c[2] * torch.cos(d[1] + w[1]) - c[1] * torch.sin(
          d[0] + w[0]
      ) * torch.sin(d[1] + w[1])
      assert torch.allclose(res, expected, atol=tol)
  
      # test gradients
      res.backward()
      grad = (w.grad, c.grad)
  
      expected_w = torch.tensor(
          [
              -c[1] * torch.cos(d[0] + w[0]) * torch.sin(d[1] + w[1]),
              -c[1] * torch.cos(d[1] + w[1]) * torch.sin(d[0] + w[0])
              - c[2] * torch.sin(d[1] + w[1]),
          ]
      )
      expected_c = torch.tensor(
          [
              0,
              -torch.sin(d[0] + w[0]) * torch.sin(d[1] + w[1]),
              torch.cos(d[1] + w[1]),
          ]
      )
      assert torch.allclose(grad[0], expected_w, atol=tol)
      assert torch.allclose(grad[1], expected_c, atol=tol)
  
      # test second-order derivatives
      if (
          diff_method in ("parameter-shift", "backprop")
          and max_diff == 2
      ):
          hessians = torch.autograd.functional.hessian(
              circuit, (d, w, c)
          )
  
          grad2_c = hessians[2][2]
          assert torch.allclose(
              grad2_c,
              torch.zeros([3, 3], dtype=torch.float64),
              atol=tol,
          )
  
          grad2_w_c = hessians[1][2]
          expected = torch.tensor(
              [
                  [
                      0,
                      -torch.cos(d[0] + w[0]) * torch.sin(d[1] + w[1]),
                      0,
                  ],
                  [
                      0,
                      -torch.cos(d[1] + w[1]) * torch.sin(d[0] + w[0]),
                      -torch.sin(d[1] + w[1]),
                  ],
              ]
          )
          assert torch.allclose(grad2_w_c, expected, atol=tol)
  
imports: import numpy
  import pytest
  import pennylane
  from pennylane import qnode
  from pennylane.devices import DefaultQubit
label_desc: Use image node_id 3 for calling the TestTapeExpansion obj's underlying member method code with example usage: obj.test_hamiltonian_expansion_analytic(dev, diff_method, grad_on_execution, max_diff, device_vjp, tol) and returns: qml