code: def mitigate_with_zne(
      tape: QuantumTape,
      scale_factors: Sequence[float],
      folding: callable,
      extrapolate: callable,
      folding_kwargs: Optional[Dict[str, Any]] = None,
      extrapolate_kwargs: Optional[Dict[str, Any]] = None,
      reps_per_factor=1,
  ) -> (Sequence[QuantumTape], Callable):
      r"""Mitigate an input circuit using zero-noise extrapolation.
  
      Error mitigation is a precursor to error correction and is compatible with near-term quantum
      devices. It aims to lower the impact of noise when evaluating a circuit on a quantum device by
      evaluating multiple variations of the circuit and post-processing the results into a
      noise-reduced estimate. This transform implements the zero-noise extrapolation (ZNE) method
      originally introduced by
      `Temme et al. <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.180509>`__ and
      `Li et al. <https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.021050>`__.
  
      Details on the functions passed to the ``folding`` and ``extrapolate`` arguments of this
      transform can be found in the usage details. This transform is compatible with functionality
      from the `Mitiq <https://mitiq.readthedocs.io/en/stable/>`__ package (version 0.11.0 and above),
      see the example and usage details for further information.
  
      Args:
          tape (QNode or QuantumTape): the quantum circuit to be error-mitigated
          scale_factors (Sequence[float]): the range of noise scale factors used
          folding (callable): a function that returns a folded circuit for a specified scale factor
          extrapolate (callable): a function that returns an extrapolated result when provided a
              range of scale factors and corresponding results
          folding_kwargs (dict): optional keyword arguments passed to the ``folding`` function
          extrapolate_kwargs (dict): optional keyword arguments passed to the ``extrapolate`` function
          reps_per_factor (int): Number of circuits generated for each scale factor. Useful when the
              folding function is stochastic.
  
      Returns:
          qnode (QNode) or tuple[List[.QuantumTape], function]:
  
          The transformed circuit as described in :func:`qml.transform <pennylane.transform>`. Executing this circuit
          will provide the mitigated results in the form of a tensor of a tensor, a tuple, or a nested tuple depending
          upon the nesting structure of measurements in the original circuit.
  
      **Example:**
  
      We first create a noisy device using ``default.mixed`` by adding :class:`~.AmplitudeDamping` to
      each gate of circuits executed on the device using the :func:`~.transforms.insert` transform:
  
      .. code-block:: python3
  
          import pennylane as qml
  
          noise_strength = 0.05
  
          dev = qml.device("default.mixed", wires=2)
          dev = qml.transforms.insert(qml.AmplitudeDamping, noise_strength)(dev)
  
      We can now set up a mitigated QNode by passing a ``folding`` and ``extrapolate`` function. PennyLane provides native
      functions :func:`~.pennylane.transforms.fold_global` and :func:`~.pennylane.transforms.poly_extrapolate` or :func:`~.pennylane.transforms.richardson_extrapolate` that
      allow for differentiating through them. Custom functions, as well as functionalities from the `Mitiq <https://mitiq.readthedocs.io/en/stable/>`__ package
      are supported as well (see usage details below).
  
      .. code-block:: python3
  
          from functools import partial
          from pennylane import numpy as np
          from pennylane import qnode
  
          from pennylane.transforms import fold_global, poly_extrapolate
  
          n_wires = 2
          n_layers = 2
  
          shapes = qml.SimplifiedTwoDesign.shape(n_wires, n_layers)
          np.random.seed(0)
          w1, w2 = [np.random.random(s) for s in shapes]
  
          @partial(qml.transforms.mitigate_with_zne, [1., 2., 3.], fold_global, poly_extrapolate, extrapolate_kwargs={'order': 2})
          @qnode(dev)
          def circuit(w1, w2):
              qml.SimplifiedTwoDesign(w1, w2, wires=range(2))
              return qml.expval(qml.PauliZ(0))
  
      Executions of ``circuit`` will now be mitigated:
  
      >>> circuit(w1, w2)
      0.19113067083636542
  
      The unmitigated circuit result is ``0.33652776`` while the ideal circuit result is
      ``0.23688169`` and we can hence see that mitigation has helped reduce our estimation error.
  
      This mitigated qnode can be differentiated like any other qnode.
  
      >>> qml.grad(circuit)(w1, w2)
      (array([-0.89319941,  0.37949841]),
       array([[[-7.04121596e-01,  3.00073104e-01]],
              [[-6.41155176e-01,  8.32667268e-17]]]))
  
      .. details::
          :title: Usage Details
  
          **Theoretical details**
  
          A summary of ZNE can be found in `LaRose et al. <https://arxiv.org/abs/2009.04417>`__. The
          method works by assuming that the amount of noise present when a circuit is run on a
          noisy device is enumerated by a parameter :math:`\gamma`. Suppose we have an input circuit
          that experiences an amount of noise equal to :math:`\gamma = \gamma_{0}` when executed.
          Ideally, we would like to evaluate the result of the circuit in the :math:`\gamma = 0`
          noise-free setting.
  
          To do this, we create a family of equivalent circuits whose ideal noise-free value is the
          same as our input circuit. However, when run on a noisy device, each circuit experiences
          a noise equal to :math:`\gamma = s \gamma_{0}` for some scale factor :math:`s`. By
          evaluating the noisy outputs of each circuit, we can extrapolate to :math:`s=0` to estimate
          the result of running a noise-free circuit.
  
          A key element of ZNE is the ability to run equivalent circuits for a range of scale factors
          :math:`s`. When the noise present in a circuit scales with the number of gates, :math:`s`
          can be varied using `unitary folding <https://ieeexplore.ieee.org/document/9259940>`__.
          Unitary folding works by noticing that a unitary :math:`U` is equivalent to
          :math:`U U^{\dagger} U`. This type of transform can be applied to individual gates in the
          circuit or to the whole circuit. When no folding occurs, the scale factor is
          :math:`s=1` and we are running our input circuit. On the other hand, when each gate has been
          folded once, we have tripled the amount of noise in the circuit so that :math:`s=3`. For
          :math:`s \geq 3`, each gate in the circuit will be folded more than once. A typical choice
          of scale parameters is :math:`(1, 2, 3)`.
  
          **Unitary folding**
  
          This transform applies ZNE to an input circuit using the unitary folding approach. It
          requires a callable to be passed as the ``folding`` argument with signature
  
          .. code-block:: python
  
              fn(circuit, scale_factor, **folding_kwargs)
  
          where
  
          - ``circuit`` is a quantum tape,
  
          - ``scale_factor`` is a float, and
  
          - ``folding_kwargs`` are optional keyword arguments.
  
          The output of the function should be the folded circuit as a quantum tape.
          Folding functionality is available from the
          `Mitiq <https://mitiq.readthedocs.io/en/stable/>`__ package (version 0.11.0 and above)
          in the
          `zne.scaling.folding <https://mitiq.readthedocs.io/en/stable/apidoc.html#module-mitiq.zne.scaling.folding>`__
          module.
  
          .. warning::
  
              Calculating the gradient of mitigated circuits is not supported when using the Mitiq
              package as a backend for folding or extrapolation.
  
          **Extrapolation**
  
          This transform also requires a callable to be passed to the ``extrapolate`` argument that
          returns the extrapolated value(s). Its function should be
  
          .. code-block:: python
  
              fn(scale_factors, results, **extrapolate_kwargs)
  
          where
  
          - ``scale_factors`` are the ZNE scale factors,
  
          - ``results`` are the execution results of the circuit at the specified scale
            factors of shape ``(len(scale_factors), len(qnode_returns))``, and
  
          - ``extrapolate_kwargs`` are optional keyword arguments.
  
          The output of the extrapolate ``fn`` should be a flat array of
          length ``len(qnode_returns)``.
  
          Extrapolation functionality is available using ``extrapolate``
          methods of the factories in the
          `mitiq.zne.inference <https://mitiq.readthedocs.io/en/stable/apidoc.html#module-mitiq.zne.inference>`__
          module.
      """
      folding_kwargs = folding_kwargs or {}
      extrapolate_kwargs = extrapolate_kwargs or {}
  
      tape = tape.expand(
          stop_at=lambda op: not isinstance(op, QuantumScript)
      )
      script_removed = QuantumScript(tape.operations[tape.num_preps :])
  
      tapes = [
          [
              folding(script_removed, s, **folding_kwargs)
              for _ in range(reps_per_factor)
          ]
          for s in scale_factors
      ]
  
      tapes = [
          tape_ for tapes_ in tapes for tape_ in tapes_
      ]  # flattens nested list
  
      # if folding was a batch transform, ignore the processing function
      if (
          isinstance(tapes[0], tuple)
          and isinstance(tapes[0][0], list)
          and callable(tapes[0][1])
      ):
          tapes = [t[0] for t, _ in tapes]
  
      prep_ops = tape.operations[: tape.num_preps]
      out_tapes = [
          QuantumScript(prep_ops + tape_.operations, tape.measurements)
          for tape_ in tapes
      ]
  
      def processing_fn(results):
          """Maps from input tape executions to an error-mitigated estimate"""
          for i, tape in enumerate(out_tapes):
              # stack the results if there are multiple measurements
              # this will not create ragged arrays since only expval measurements are allowed
              if len(tape.observables) > 1:
                  results[i] = qml.math.stack(results[i])
  
          # Averaging over reps_per_factor repetitions
          results_flattened = []
          for i in range(0, len(results), reps_per_factor):
              # The stacking ensures the right interface is used
              # averaging over axis=0 is critical because the qnode may have multiple outputs
              results_flattened.append(
                  mean(
                      qml.math.stack(results[i : i + reps_per_factor]),
                      axis=0,
                  )
              )
  
          extrapolated = extrapolate(
              scale_factors, results_flattened, **extrapolate_kwargs
          )
  
          extrapolated = (
              extrapolated[0]
              if shape(extrapolated) == (1,)
              else extrapolated
          )
  
          # unstack the results in the case of multiple measurements
          return (
              extrapolated
              if shape(extrapolated) == ()
              else tuple(qml.math.unstack(extrapolated))
          )
  
      return out_tapes, processing_fn
  
file_path: pennylane/pennylane/transforms/mitigate.py
imports: from copy import copy
  from typing import Any, Dict, Optional, Sequence, Callable
  from pennylane import apply, adjoint
  from pennylane.math import mean, shape, round
  from pennylane.queuing import AnnotatedQueue
  from pennylane.tape import QuantumTape, QuantumScript
  from pennylane.transforms import transform
  import pennylane
label_desc: Use image node_id 7 for calling a global function with example usage: mitigate_with_zne(tape, scale_factors, folding, extrapolate, folding_kwargs, extrapolate_kwargs, reps_per_factor) and returns: out_tapes,  processing_fn, unknown