Quantum Fourier Transform: Qudit gates

Here we move in to generalize the basic Qubit gates to Qudit gates.

We take some common gates like

Hadamard gate : Hadamard gate is used widely in Qubit literature. Hadamard gate is given by

$H= \frac{1}{\sqrt{2}}\begin{bmatrix} 1&1\\ 1&-1\end{bmatrix}$

This we have noticed that is the $F_2$ . Hence we take the $F_d$ as the generalized Qudit gate for the Hadamard gate.

NOT Gate : For a Qubit system with canonical states $\{|0>,|1>\}$ the not gate acts on a possible state $|u> = a|0>+b|1>$ as

$X|u>=X(a|0>+b|1>)=a|1>+b|0>$

We generalize this gate to its Qudit counterpart as $X_d|a>=|a+1>$ where $|a>$ is a canonical qudit state and the operation + is the ring operation over $Z_d=\{0,1,\dots,d-1\}$ ."1" is the additive identity of $Z_d$

Now suppose $x\inZ_d$ then we have

$X_d|x> = |x+1>$
$X_d^2|x>=X_d|x+1> = |x+2>$

Like wise we get

$X_d^n|x>=|x+n>$

1 comment:

Ulyses Rico ReaMay 25, 2017 at 11:08 AM
Thank you! I'm just trying to understand Quantum computing more. Your post is nice. Couls you recommend me some book which you consider is easy to understand? I'm working with "Quantum computing explained" but when it arrives to some topics such as quantum algorithms I get lost.

Quantum Fourier Transform

Monday, April 20, 2009

Qudit gates

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