NaiveApproxBackend
^^^^^^^^^^^^^^^^^^

Like the :ref:`NaiveBackend`, the :code:`NaiveApproxBackend` is a strong simulation backend that can compute a
single output probability amplitude at a time, by computing the permanent of a :math:`n \times n` matrix.
However, instead of computing exactly the permanent, the :code:`NaiveApproxBackend` uses the Gurvit estimate algorithm
to approximate it with a 99% confidence interval (see :cite:t:`gurvits2002`).

It is very efficient to compute very precise output states, but not to compute the whole distribution, and it can be used
with more modes and photons than the :ref:`NaiveBackend` at the cost of losing precision on the result.

Thus, using it is not recommended in :ref:`Simulator` (except when using :meth:`probability()`) or :ref:`Processor`,
but it is well suited for applications where only a few output probabilities matter with many photons.
If the whole or most of the computational space is needed, other backends like :ref:`SLOSBackend` are more suited.

This backend is available in :ref:`Processor` by using the name :code:`"NaiveApprox"`.

Unlike other backends, this backend needs a number of iterations to use to estimate the permanent.
Also, in addition to the generic backend methods, this backend offers means to get a 99% confidence interval on the probability
or a 99% sure error bound on the amplitude.

>>> import perceval as pcvl
>>> circuit_size = 60
>>> n_photons = 30
>>> backend = pcvl.NaiveApproxBackend(100_000_000)  # Number of iterations; higher values reduce the error bound
>>> backend.set_circuit(pcvl.Unitary(pcvl.Matrix.random_unitary(circuit_size)))
>>> input_state = pcvl.BasicState([1]*n_photons + [0]*(circuit_size-n_photons))
>>> backend.set_input_state(input_state)
>>> interval = backend.probability_confidence_interval(BasicState([1]*n_photons + [0]*(circuit_size-n_photons)))
>>> print(f"Probability in {interval}")
Probability in [6.051670221391749e-20, 1.5297683283662674e-19]

.. autoclass:: perceval.backends._naive_approx.NaiveApproxBackend
   :members:
   :inherited-members: