code: def _compute_fidelity_vjp0(dm0, dm1, grad_out):
      """
      Compute the VJP of fidelity with respect to the first density matrix
      """
      # sqrt of sigma
      sqrt_dm1 = qml.math.sqrt_matrix(dm1)
  
      # eigendecomposition of sqrt(sigma) * rho * sqrt(sigma)
      evs0, u0 = qml.math.linalg.eigh(sqrt_dm1 @ dm0 @ sqrt_dm1)
      evs0 = qml.math.real(evs0)
      evs0 = qml.math.where(evs0 > 1e-15, evs0, 1e-15)
      evs0 = qml.math.cast_like(evs0, sqrt_dm1)
  
      if (
          len(qml.math.shape(dm0)) == 2
          and len(qml.math.shape(dm1)) == 2
      ):
          u0_dag = qml.math.transpose(qml.math.conj(u0))
          grad_dm0 = (
              sqrt_dm1
              @ u0
              @ (1 / qml.math.sqrt(evs0)[..., None] * u0_dag)
              @ sqrt_dm1
          )
  
          # torch and tensorflow use the Wirtinger derivative which is a different convention
          # than the one autograd and jax use for complex differentiation
          if qml.math.get_interface(dm0) in ["torch", "tensorflow"]:
              grad_dm0 = (
                  qml.math.sum(qml.math.sqrt(evs0), -1) * grad_dm0
              )
          else:
              grad_dm0 = qml.math.sum(
                  qml.math.sqrt(evs0), -1
              ) * qml.math.transpose(grad_dm0)
  
          res = grad_dm0 * qml.math.cast_like(grad_out, grad_dm0)
          return res
  
      # broadcasting case
      u0_dag = qml.math.transpose(qml.math.conj(u0), (0, 2, 1))
      grad_dm0 = (
          sqrt_dm1
          @ u0
          @ (1 / qml.math.sqrt(evs0)[..., None] * u0_dag)
          @ sqrt_dm1
      )
  
      # torch and tensorflow use the Wirtinger derivative which is a different convention
      # than the one autograd and jax use for complex differentiation
      if qml.math.get_interface(dm0) in ["torch", "tensorflow"]:
          grad_dm0 = (
              qml.math.sum(qml.math.sqrt(evs0), -1)[:, None, None]
              * grad_dm0
          )
      else:
          grad_dm0 = qml.math.sum(qml.math.sqrt(evs0), -1)[
              :, None, None
          ] * qml.math.transpose(grad_dm0, (0, 2, 1))
  
      return (
          grad_dm0
          * qml.math.cast_like(grad_out, grad_dm0)[:, None, None]
      )
  
file_path: pennylane/pennylane/math/fidelity.py
imports: from functools import lru_cache
  import autograd
  import autoray
  import pennylane
  from .utils import cast
  from .quantum import _check_density_matrix, _check_state_vector
label_desc: Use image node_id 5 for calling a global function with example usage: _compute_fidelity_vjp0(dm0, dm1, grad_out) and returns: unknown, res