code: def _group_commutator_decompose(matrix, tol=1e-5):
      r"""Performs a group commutator decomposition :math:`U = V' \times W' \times V'^{\dagger} \times W'^{\dagger}`
      as given in the Section 4.1 of `arXiv:0505030 <https://arxiv.org/abs/quant-ph/0505030>`_.
      """
      # Use the quaternion form to get the rotation axis and angle on the Bloch sphere,
      # while using clipping for dealing with floating point precision errors.
      quaternion = _quaternion_transform(matrix)
      theta, axis = (
          2 * qml.math.arccos(qml.math.clip(quaternion[0], -1.0, 1.0)),
          quaternion[1:],
      )
  
      # Early return for the case where matrix is I or -I, where I is Identity.
      if qml.math.allclose(axis, 0.0, atol=tol) and qml.math.isclose(
          theta % math.pi, 0.0, atol=tol
      ):
          return qml.math.eye(2, dtype=complex), qml.math.eye(
              2, dtype=complex
          )
  
      # The angle phi comes from the Eq. 10 in the Solovay-Kitaev algorithm paper (arXiv:0505030).
      phi = 2.0 * qml.math.arcsin(
          qml.math.sqrt(
              qml.math.sqrt((0.5 - 0.5 * qml.math.cos(theta / 2)))
          )
      )
  
      # Begin decomposition by computing the rotation operations V and W.
      v = qml.RX(phi, [0])
      w = (
          qml.RY(2 * math.pi - phi, [0])
          if axis[2] > 0
          else qml.RY(phi, [0])
      )
  
      # Get the similarity transormation matrices S and S.adjoint().
      ud = qml.math.linalg.eig(matrix)[1]
      vwd = qml.math.linalg.eig(
          qml.matrix(v @ w @ v.adjoint() @ w.adjoint())
      )[1]
      s = ud @ qml.math.conj(qml.math.transpose(vwd))
      sdg = vwd @ qml.math.conj(qml.math.transpose(ud))
  
      # Get the required matrices V' and W'.
      v_hat = s @ v.matrix() @ sdg
      w_hat = s @ w.matrix() @ sdg
  
      return w_hat, v_hat
  
file_path: pennylane/pennylane/ops/op_math/decompositions/solovay_kitaev.py
imports: import math
  import warnings
  from functools import lru_cache
  from scipy.spatial import KDTree
  import pennylane
  from pennylane.queuing import QueuingManager
  from pennylane.tape import QuantumScript
label_desc: Use image node_id 5 for calling a global function with example usage: _group_commutator_decompose(matrix, tol) and returns: w_hat,  v_hat, qml,  qml