code: def _approximate_set(basis_gates, max_length=10):
      r"""Builds an approximate unitary set required for the `Solovay-Kitaev algorithm <https://arxiv.org/abs/quant-ph/0505030>`_.
  
      Args:
          basis_gates (list(str)): Basis set to be used for Solovay-Kitaev decomposition build using
              following terms, ``['X', 'Y', 'Z', 'H', 'T', 'T*', 'S', 'S*']``, where `*` refers
              to the gate adjoint.
          max_length (int): Maximum expansion length of Clifford+T sequences in the approximation set. Default is `10`
  
      Returns:
          Tuple(list[list[~pennylane.operation.Operation]], list[TensorLike], list[float], list[TensorLike]): A tuple containing the list of
          Clifford+T sequences that will be used for approximating a matrix in the base case of recursive implementation of
          Solovay-Kitaev algorithm, with their corresponding SU(2) representations, global phases, and quaternion representations.
      """
      # Defining Clifford+T basis
      _CLIFFORD_T_BASIS = {
          "I": qml.Identity(0),
          "X": qml.PauliX(0),
          "Y": qml.PauliY(0),
          "Z": qml.PauliZ(0),
          "H": qml.Hadamard(0),
          "T": qml.T(0),
          "T*": qml.adjoint(qml.T(0)),
          "S": qml.S(0),
          "S*": qml.adjoint(qml.S(0)),
      }
      # Maintains the basis gates
      basis = [_CLIFFORD_T_BASIS[gate.upper()] for gate in basis_gates]
  
      # Get the SU(2) data for the gates in basis set
      basis_mat, basis_gph = {}, {}
      for gate in basis:
          su2_mat, su2_gph = _SU2_transform(gate.matrix())
          basis_mat.update({gate: su2_mat})
          basis_gph.update({gate: su2_gph})
  
      # Maintains a trie-like structure for each depth
      gtrie_ids = [[[gate] for gate in basis]]
      gtrie_mat = [list(basis_mat.values())]
      gtrie_gph = [list(basis_gph.values())]
  
      # Maintains the approximate set for gates' names, SU(2)s, global phases and quaternions
      approx_set_ids = list(gtrie_ids[0])
      approx_set_mat = list(gtrie_mat[0])
      approx_set_gph = list(gtrie_gph[0])
      approx_set_qat = [
          _quaternion_transform(mat) for mat in approx_set_mat
      ]
  
      # We will perform a breadth-first search (BFS) style set building for the set
      for depth in range(max_length - 1):
          # Add the containers for next depth while we explore the current
          gtrie_id, gtrie_mt, gtrie_gp = [], [], []
          for node, su2m, gphase in zip(
              gtrie_ids[depth], gtrie_mat[depth], gtrie_gph[depth]
          ):
              # Get the last operation in the current node
              last_op = (
                  qml.adjoint(node[-1], lazy=False) if node else None
              )
  
              # Now attempt extending the current node for each basis gate
              for op in basis:
                  # If basis gate is adjoint of last op in the node, skip.
                  if qml.equal(op, last_op):
                      continue
  
                  # Extend and check if the node already exists in the approximate set.
                  su2_gp = basis_gph[op] + gphase
                  su2_op = (-1.0) ** bool(su2_gp >= math.pi) * (
                      basis_mat[op] @ su2m
                  )
                  exists, quaternion = _contains_SU2(
                      su2_op, approx_set_qat
                  )
                  if not exists:
                      approx_set_ids.append(node + [op])
                      approx_set_mat.append(su2_op)
                      approx_set_qat.append(quaternion)
  
                      # Add to the containers for next depth
                      gtrie_id.append(node + [op])
                      gtrie_mt.append(su2_op)
  
                      # Add the global phase data
                      global_phase = qml.math.mod(su2_gp, math.pi)
                      approx_set_gph.append(global_phase)
                      gtrie_gp.append(global_phase)
  
          # Add to the next depth for next iteration
          gtrie_ids.append(gtrie_id)
          gtrie_mat.append(gtrie_mt)
          gtrie_gph.append(gtrie_gp)
  
      return (
          approx_set_ids,
          approx_set_mat,
          approx_set_gph,
          approx_set_qat,
      )
  
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 4 for calling a global function with example usage: _approximate_set(basis_gates, max_length) and returns: approx_set_ids,  approx_set_mat,  approx_set_gph,  approx_set_qat