code: def clifford_t_decomposition(
      tape: QuantumTape,
      epsilon=1e-4,
      max_expansion=6,
      method="sk",
      **method_kwargs,
  ) -> (Sequence[QuantumTape], Callable):
      r"""Decomposes a circuit into the Clifford+T basis.
  
      This method first decomposes the gate operations to a basis comprised of Clifford, :class:`~.T`, :class:`~.RZ` and
      :class:`~.GlobalPhase` operations (and their adjoints). The Clifford gates include the following PennyLane operations:
  
      - Single qubit gates - :class:`~.Identity`, :class:`~.PauliX`, :class:`~.PauliY`, :class:`~.PauliZ`,
        :class:`~.SX`, :class:`~.S`, and :class:`~.Hadamard`.
      - Two qubit gates - :class:`~.CNOT`, :class:`~.CY`, :class:`~.CZ`, :class:`~.SWAP`, and :class:`~.ISWAP`.
  
      Then, the leftover single qubit :class:`~.RZ` operations are approximated in the Clifford+T basis with
      :math:`\epsilon > 0` error. By default, we use the Solovay-Kitaev algorithm described in
      `Dawson and Nielsen (2005) <https://arxiv.org/abs/quant-ph/0505030>`_ for this.
  
      Args:
          tape (QNode or QuantumTape or Callable): The quantum circuit to be decomposed.
          epsilon (float): The maximum permissible operator norm error of the complete circuit decomposition. Defaults to ``0.0001``.
          max_expansion (int): The depth to be used for tape expansion before manual decomposition to Clifford+T basis is applied.
          method (str): Method to be used for Clifford+T decomposition. Default value is ``"sk"`` for Solovay-Kitaev.
          **method_kwargs: Keyword argument to pass options for the ``method`` used for decompositions.
  
      Returns:
          qnode (QNode) or quantum function (Callable) or tuple[List[QuantumTape], function]: The transformed circuit as described
          in the :func:`qml.transform <pennylane.transform>`.
  
      **Keyword Arguments**
  
      - Solovay-Kitaev decomposition --
          **max_depth** (int), **basis_set** (list[str]), **basis_length** (int) -- arguments for the ``"sk"`` method,
          where the decomposition is performed using the :func:`~.sk_decomposition` method.
  
      Raises:
          ValueError: If a gate operation does not have a decomposition when required.
          NotImplementedError: If chosen decomposition ``method`` is not supported.
  
      .. seealso:: :func:`~.sk_decomposition` for Solovay-Kitaev decomposition.
  
      **Example**
  
      .. code-block:: python3
  
          @qml.qnode(qml.device("default.qubit"))
          def circuit(x, y):
              qml.RX(x, 0)
              qml.CNOT([0, 1])
              qml.RY(y, 0)
              return qml.expval(qml.PauliZ(0))
  
          x, y = 1.1, 2.2
          decomposed_circuit = qml.transforms.clifford_t_decomposition(circuit)
          result = circuit(x, y)
          approx = decomposed_circuit(x, y)
  
      >>> qml.math.allclose(result, approx, atol=1e-4)
      True
      """
      with QueuingManager.stop_recording():
          # Build the basis set and the pipeline for intial compilation pass
          basis_set = [
              op.__name__ for op in _PARAMETER_GATES + _CLIFFORD_T_GATES
          ]
          pipelines = [
              remove_barrier,
              commute_controlled,
              cancel_inverses,
              merge_rotations,
          ]
  
          # Compile the tape according to depth provided by the user and expand it
          [compiled_tape], _ = qml.compile(
              tape,
              pipelines,
              basis_set=basis_set,
              expand_depth=max_expansion,
          )
  
          # Now iterate over the expanded tape operations
          decomp_ops, gphase_ops = [], []
          for op in compiled_tape.operations:
              # Check whether operation is to be skipped
              if isinstance(op, _SKIP_OP_TYPES):
                  decomp_ops.append(op)
  
              # Check whether the operation is a global phase
              elif isinstance(op, qml.GlobalPhase):
                  gphase_ops.append(op)
  
              # Check whether the operation is a Clifford or a T-gate
              elif op.name in basis_set and check_clifford_t(op):
                  if op.num_params:
                      decomp_ops.extend(_rot_decompose(op))
                  else:
                      decomp_ops.append(op)
  
              # Decompose and then iteratively go deeper via DFS
              else:
                  # Single qubit unitary decomposition with ZXZ rotations
                  if op.num_wires == 1:
                      if op.name in basis_set:
                          d_ops = _rot_decompose(op)
                      else:
                          d_ops, g_op = _one_qubit_decompose(op)
                          gphase_ops.append(g_op)
                      decomp_ops.extend(d_ops)
  
                  # Two qubit unitary decomposition with SU(4) rotations
                  elif op.num_wires == 2:
                      d_ops = _two_qubit_decompose(op)
                      decomp_ops.extend(d_ops)
  
                  # For special multi-qubit gates and ones constructed from matrix
                  else:
                      try:
                          # Attempt decomposing the operation
                          md_ops = op.decomposition()
                          idx = 0  # might not be fast but at least is not recursive
                          while idx < len(md_ops):
                              md_op = md_ops[idx]
                              if (
                                  md_op.name not in basis_set
                                  or not check_clifford_t(md_op)
                              ):
                                  # For the gates acting on one qubit
                                  if len(md_op.wires) == 1:
                                      if (
                                          md_op.name in basis_set
                                      ):  # For known recipe
                                          d_ops = _rot_decompose(md_op)
                                      else:  # Resort to decomposing manually
                                          (
                                              d_ops,
                                              g_op,
                                          ) = _one_qubit_decompose(
                                              md_op
                                          )
                                          gphase_ops.append(g_op)
  
                                  # For the gates acting on two qubits
                                  elif len(md_op.wires) == 2:
                                      # Resort to decomposing manually
                                      d_ops = _two_qubit_decompose(
                                          md_op
                                      )
  
                                  # Final resort (should not enter in an ideal situtation)
                                  else:
                                      d_ops = md_op.decomposition()
  
                                  # Expand the list and iterate over
                                  del md_ops[idx]
                                  md_ops[idx:idx] = d_ops
                              idx += 1
  
                          decomp_ops.extend(md_ops)
  
                      # If we don't know how to decompose the operation
                      except Exception as exc:
                          raise ValueError(
                              f"Cannot unroll {op} into the Clifford+T basis as no rule exists for its decomposition"
                          ) from exc
  
          # Merge RZ rotations together
          merged_ops, number_ops = _merge_param_gates(
              decomp_ops, merge_ops=["RZ"]
          )
  
          # Squeeze global phases into a single global phase
          new_operations = _fuse_global_phases(merged_ops + gphase_ops)
  
          # Compute the per-gate epsilon value
          epsilon /= number_ops or 1
  
          # Every decomposition implementation should have the following shape:
          # def decompose_fn(op: Operator, epsilon: float, **method_kwargs) -> List[Operator]
          # note: the last operator in the decomposition must be a GlobalPhase
  
          # Build the approximation set for Solovay-Kitaev decomposition
          if method == "sk":
              decompose_fn = sk_decomposition
  
          else:
              raise NotImplementedError(
                  f"Currently we only support Solovay-Kitaev ('sk') decompostion, got {method}"
              )
  
          decomp_ops = []
          phase = new_operations.pop().data[0]
          for op in new_operations:
              if isinstance(op, qml.RZ):
                  clifford_ops = decompose_fn(
                      op, epsilon, **method_kwargs
                  )
                  phase += qml.math.convert_like(
                      clifford_ops.pop().data[0], phase
                  )
                  decomp_ops.extend(clifford_ops)
              else:
                  decomp_ops.append(op)
  
          # check if phase is non-zero for non jax-jit cases
          if qml.math.is_abstract(phase) or not qml.math.allclose(
              phase, 0.0
          ):
              decomp_ops.append(qml.GlobalPhase(phase))
  
      # Construct a new tape with the expanded set of operations
      new_tape = type(tape)(
          decomp_ops, compiled_tape.measurements, shots=tape.shots
      )
  
      # Perform a final attempt of simplification before return
      [new_tape], _ = cancel_inverses(new_tape)
  
      def null_postprocessing(results):
          """A postprocesing function returned by a transform that only converts the batch of results
          into a result for a single ``QuantumTape``.
          """
          return results[0]
  
      return [new_tape], null_postprocessing
  
file_path: pennylane/pennylane/transforms/decompositions/clifford_t_transform.py
imports: import math
  import warnings
  from itertools import product
  from typing import Sequence, Callable
  import pennylane
  from pennylane.queuing import QueuingManager
  from pennylane.transforms.core import transform
  from pennylane.tape import QuantumTape
  from pennylane.transforms.optimization import cancel_inverses, commute_controlled, merge_rotations, remove_barrier
  from pennylane.transforms.optimization.optimization_utils import find_next_gate, _fuse_global_phases
  from pennylane.ops.op_math.decompositions.solovay_kitaev import sk_decomposition
label_desc: Use image node_id 8 for calling a global function with example usage: clifford_t_decomposition(tape, epsilon, max_expansion, method) and returns: list,  null_postprocessing, results