toqito.matrices.fourier

Generates a Fourier matrix.

Module Contents

toqito.matrices.fourier.fourier(dim)

Generate the Fourier transform matrix [@WikiDFT].

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

The Fourier matrix is defined as:

[

W_N = frac{1}{sqrt{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^{(N-1)(N-1)}

end{pmatrix}

]

Examples

The Fourier matrix generated from (d = 3) yields the following matrix:

[

W_3 = frac{1}{sqrt{3}} begin{pmatrix}

1 & 1 & 1 \ 1 & omega & omega^2 \ 1 & omega^2 & omega^4

end{pmatrix}

]

```python exec=”1” source=”above” from toqito.matrices import fourier

print(fourier(3)) ```

Parameters:

dim (int) – The size of the Fourier matrix.

Returns:

The Fourier matrix of dimension dim.

Return type:

numpy.ndarray