channels.pauli_channel
======================

.. py:module:: channels.pauli_channel

.. autoapi-nested-parse::

   Generates and applies Pauli Channel to a matrix.



Functions
---------

.. autoapisummary::

   channels.pauli_channel.pauli_channel


Module Contents
---------------

.. py:function:: pauli_channel(prob, return_kraus_ops = False, input_mat = None)

   Generate and apply a Pauli channel to a matrix.

   Generates the Choi matrix of a Pauli channel with given probabilities and optionally applies it
   to an input matrix. The Pauli channel is defined by the set of Pauli operators weighted by
   the probability vector. For a given probability vector :math:`(p_0, \ldots, p_{4^q -1 })`, the
   channel is defined as shown below. Where, $q$ is the number of qubits.

   .. math::
      \Phi(\rho) = \sum_{i=0}^{4^q - 1} p_i P_i \rho P_i^*

   where :math:`P_i` are Pauli operators generated by a lexographically increasing sequence of pauli operators of
   length strictly equal to :math:`q`, and :math:`p_i` is the corresponding probability of that operator.
   For example, when :math:`q = 2`,

   :math:`P_{0} = I \otimes I`, :math:`P_{1} = I \otimes X`, :math:`P_{2} = I \otimes Y`, :math:`P_{3} = I \otimes Z`,
   :math:`P_{4} = X \otimes I`, :math:`P_{5} = X \otimes Y , \ldots P_{15} = Z \otimes Z`

   If :code:`prob` is a scalar, it generates a random :code:`prob`-qubit Pauli channel.
   The length of the probability vector (if provided) must be :math:`4^q` for some
   integer :math:`q` (number of qubits).

   .. rubric:: Examples

   Generate a random single-qubit Pauli channel:

   .. jupyter-execute::

    from toqito.channels import pauli_channel

    pauli_channel(prob=1)

   Apply a specific two-qubit Pauli channel to an input matrix:

   .. jupyter-execute::

    import numpy as np
    from toqito.channels import pauli_channel

    _, output = pauli_channel(
        prob=np.array([0.1, 0.2, 0.3, 0.4]), input_mat=np.eye(2)
    )
    print(output)

   .. rubric:: References

   .. footbibliography::

   :param prob: Probability vector for Pauli operators. If scalar, generates random probabilities
            for :math:`q =` `prob` qubits. The probabilities correspond to Pauli operators in
            lexographical order of length strictly equal to :math:`q` ,when `prob` is a vector.
   :param return_kraus_ops: Flag to return Kraus operators. Default is ``False``.
   :param input_mat: Optional input matrix to apply the channel to. Default is ``None``.
   :raises ValueError: If probabilities are negative or don't sum to 1.
   :raises ValueError: If length of probability vector is not ``4^q`` for some integer ``q``.
   :return: The Choi matrix of the channel.
        If ``input_mat`` is provided, also returns the output matrix.
        If ``return_kraus_ops`` is ``True``, returns Kraus operators as well.