Thursday, April 23, 2009

Qudit Swap Gate

We have seen that the contolled-X (C-NOT) operation in d-state systems acts as

$|c,x>\to |c,x+c>$

Then the Controlled - $X_d^H$ operation acts as

$|c,x>\to |c,x-c>$

Using these operators we can make the Qudit Swap Gate.

Now consider the following gates
Call this as Gate M1.

Call this as Gate M2

Now we can see that operation M1 is

$M_1|c,t>\to|-t,c>$

And operation M2 is

$M_2|c,t>\to|t,-c>$

Also

$M_1M_2|c,t>=M_1|t,-c>=|c,t>$

So $M_1M_2=I=M_2M_1$

Now by adding an $N_d$ operation at the end we can form the Qudit swap gates as shown

Since the inverse of Swap operation is also Swap operation we have the following implementations also

Tuesday, April 21, 2009

Controlled-U Qudit gate

Consider a Qudit Controlled-U Gate

Let the Qudit system under consideration have $\{|0>,|1>,...,|d-1>\}$ as the d-canonical states and let us consider the ring $Z_d$ be the ring of integers from zero to d-1.

Now the controlled-U operation is defined for a qudit sytem as follows

$|x,u>\rightarrow|\varphi>$ where

$|x> = \sum_{s=0}^{d-1}a_s |s>$

and

$|\varphi> = \sum_{s=0}^{d-1}a_s |s>U^s|v>$

Using this we see that the qudit version of Controlled NOT gate becomes

$|\varphi> = \sum_{s=0}^{d-1}a_s |s>X_d^s|v>=\sum_{s=0}^{d-1}a_s |s>|v+s>$

So this is in accordance with the d=2 where where had the CNOT operation doing

$|x,u>\rightarrow |x,x \oplus u>$

So the generalized CNOT operation is

$|x,u>\rightarrow |x,x + u>$

Monday, April 20, 2009

Qudit gates

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

We take some common gates like

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

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

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

Tuesday, April 14, 2009

Properties of QFT

Being similar to the DFT matrix, QFT matrix has all the properties of DFT matrix.

Some properties are listed below

• 1) , means $F_d$ is unitary matrix , like every other Quantum Operator.

• 2)

• 3) So

Now we focus on the powers of .
We have

When m+n = 0(mod d)we have on other cases the summation is is zero.So we have the

let So we have .

$N_d=\begin{bmatrix}1 & 0 & . & . & 0 & 0 & 0\\0 & 0 & . & . & 0 & 0 & 1\\0 & 0 & . & . & 0 & 1 & 0\\. & . & . & . & . & . & .\\0 & 1 & . & . & 0 & 0 & 0\end{bmatrix}$

Now $N_d^2=I$ (this is easy to prove). then

$F_d^4=(F_d^2)^2=N_d^2=I$

So we have the following results.

$N_d F_d = F_d N_d=F_d^3=F_d^4*F_d^{-1}=F_d^{-1}=F_d^H$

Note that $[N_d,F_d]=N_d F_d - F_d N_d=0$ so they commute.

Now let us consider a qudit system with d- cannonical states $|0>, |1>, ...,|d-1>$ , Consider the quantum Operator $N_d$ acting on any canonical state $|x>$, we have

$\begin{matrix}N_d|0> = |0>\\ N_d|1> = |d-1>\\ N_d|2> = |d-2>\\...\\N_d|d-1> = |1>\end{matrix}$

Consider the ring of integers $Z_d =\{0,1,2,...,d-1\}$ with normal addition and multiplication being the ring operations. Then we can view the operator $N_d$ as

$N_d|x> =|-x>$

where $|x>$ is a canonical state, where negation is the group negation operation. This is equivalent to time reversal in the DFT case.

Simplest $N_d$ matrices are given below.

$N_2=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}=I$

$N_3=\begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{bmatrix}$