Difference between revisions of "FourierMatrix"

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

Revision as of 04:14, 20 December 2014

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.

@@ Line 2: / Line 2: @@
 |name=FourierMatrix
 |cat=[[List of functions#Special_states,_vectors,_and_operators|Special states, vectors, and operators]]
-|desc=Generates the [[unitary matrix]] that implements the [[quantum Fourier transform]]
+|desc=Generates the unitary matrix that implements the quantum Fourier transform
-|upd=November 30, 2012
+|upd=November 30, 2012}}
-|v=0.50}}
+<tt>'''FourierMatrix'''</tt> is a [[List of functions|function]] that returns the unitary matrix that implements the [http://en.wikipedia.org/wiki/Quantum_Fourier_transform quantum Fourier transform]. That is, it returns the $d \times d$ matrix
-<tt>'''FourierMatrix'''</tt> is a [[List of functions|function]] that returns the [[unitary matrix]] that implements the [[quantum Fourier transform]].
+: <math>\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},</math>
+where $\omega := \exp(2\pi i/d)$ is a [http://en.wikipedia.org/wiki/Root_of_unity primitive $d$-th root of unity].
 ==Syntax==
 * <tt>F = FourierMatrix(DIM)</tt>
@@ Line 15: / Line 15: @@
 ==Examples==
 ===The qubit Fourier matrix===
-The qubit Fourier matrix is simply the usual [[Hadamard gate]]:
+The qubit Fourier matrix is simply the usual Hadamard gate:
 <syntaxhighlight>
 >> FourierMatrix(2)