:py:mod:`matrices.fourier`
==========================

.. py:module:: matrices.fourier

.. autoapi-nested-parse::

   Fourier matrix.



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


Functions
~~~~~~~~~

.. autoapisummary::

   matrices.fourier.fourier



.. py:function:: fourier(dim)

   Generate the Fourier transform matrix :cite:`WikiDFT`.

   Generates the :code:`dim`-by-:code:`dim` unitary matrix that implements the
   quantum Fourier transform.

   The Fourier matrix is defined as:

   .. math::
       W_N = \frac{1}{N}
       \begin{pmatrix}
           1 & 1 & 1 & 1 & \ldots & 1 \\
           1 & \omega & \omega^2 & \omega^3 & \ldots & \omega^{N-1} \\
           1 & \omega^2 & \omega^4 & \omega^6 & \ldots & \omega^{2(N-1)} \\
           1 & \omega^3 & \omega^6 & \omega^9 & \ldots & \omega^{3(N-1)} \\
           \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
           1 & \omega^{N-1} & \omega^{2(N-1)} & \omega^{3(N-1)} &
           \ldots & \omega^{3(N-1)}
       \end{pmatrix}

   .. rubric:: Examples

   The Fourier matrix generated from :math:`d = 3` yields the following
   matrix:

   .. math::
       W_3 = \frac{1}{3}
       \begin{pmatrix}
           1 & 1 & 1 \\
           0 & \omega & \omega^2 \\
           1 & \omega^2 & \omega^4
       \end{pmatrix}

   >>> from toqito.matrices import fourier
   >>> fourier(3)
   array([[ 0.57735027+0.j ,  0.57735027+0.j ,  0.57735027+0.j ],
          [ 0.57735027+0.j , -0.28867513+0.5j, -0.28867513-0.5j],
          [ 0.57735027+0.j , -0.28867513-0.5j, -0.28867513+0.5j]])

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param dim: The size of the Fourier matrix.
   :return: The Fourier matrix of dimension :code:`dim`.