FourierMatrix

FourierMatrix
Generates the unitary matrix that implements the quantum Fourier transform
Other toolboxes required	none
Function category	Special states, vectors, and operators

FourierMatrix is a function that returns the unitary matrix that implements the quantum Fourier transform. That is, it returns the $d \times d$ matrix \[\frac{1}{\sqrt{d}}\begin{bmatrix}1 & 1 & 1 & \cdots & 1 \\ 1 & \omega & \omega^2 & \cdots & \omega^{d-1} \\ 1 & \omega^2 & \omega^4 & \cdots & \omega^{2(d-1)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)(d-1)} \end{bmatrix},\] where $\omega := \exp(2\pi i/d)$ is a primitive d-th root of unity.

Syntax

• F = FourierMatrix(DIM)

Argument descriptions

• DIM: The dimension of the system. In other words, F will be a DIM-by-DIM matrix.

Examples

The qubit Fourier matrix

The qubit Fourier matrix is simply the usual Hadamard gate:

>> FourierMatrix(2)

ans =

   0.7071             0.7071          
   0.7071            -0.7071 + 0.0000i

The three-qubit Fourier matrix

The following line of code generates the three-qubit (i.e., DIM = 8) Fourier matrix, which can be seen here. The multiplication by sqrt(8) is just there to make the output easier to read.

>> FourierMatrix(8)*sqrt(8)

ans =

   1.0000             1.0000             1.0000             1.0000             1.0000             1.0000             1.0000             1.0000          
   1.0000             0.7071 + 0.7071i   0.0000 + 1.0000i  -0.7071 + 0.7071i  -1.0000 + 0.0000i  -0.7071 - 0.7071i  -0.0000 - 1.0000i   0.7071 - 0.7071i
   1.0000             0.0000 + 1.0000i  -1.0000 + 0.0000i  -0.0000 - 1.0000i   1.0000 - 0.0000i   0.0000 + 1.0000i  -1.0000 + 0.0000i  -0.0000 - 1.0000i
   1.0000            -0.7071 + 0.7071i  -0.0000 - 1.0000i   0.7071 + 0.7071i  -1.0000 + 0.0000i   0.7071 - 0.7071i   0.0000 + 1.0000i  -0.7071 - 0.7071i
   1.0000            -1.0000 + 0.0000i   1.0000 - 0.0000i  -1.0000 + 0.0000i   1.0000 - 0.0000i  -1.0000 + 0.0000i   1.0000 - 0.0000i  -1.0000 + 0.0000i
   1.0000            -0.7071 - 0.7071i   0.0000 + 1.0000i   0.7071 - 0.7071i  -1.0000 + 0.0000i   0.7071 + 0.7071i  -0.0000 - 1.0000i  -0.7071 + 0.7071i
   1.0000            -0.0000 - 1.0000i  -1.0000 + 0.0000i   0.0000 + 1.0000i   1.0000 - 0.0000i  -0.0000 - 1.0000i  -1.0000 + 0.0000i   0.0000 + 1.0000i
   1.0000             0.7071 - 0.7071i  -0.0000 - 1.0000i  -0.7071 - 0.7071i  -1.0000 + 0.0000i  -0.7071 + 0.7071i   0.0000 + 1.0000i   0.7071 + 0.7071i

Source code

Click here to view this function's source code on github.