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Let's assume that I'm modelling a d-levels qudit as a set of n qubits, where $d = 2^n$. I would need to map the application of quantum gates on the qudit to a set of operations of the relative qubits set.

For example, this is an example

$$ X^{1,2} \left ( \frac{1}{\sqrt{2}} , \frac{1}{\sqrt{4}} , 0 , \frac{1}{\sqrt{4}} \right ) = \left ( \frac{1}{\sqrt{2}} , 0 , \frac{1}{\sqrt{4}} , \frac{1}{\sqrt{4}} \right ) $$

In the above case I applied the X gate on the levels pair $(1, 2)$.

Given that I won't have a qudit but a set of qubits I'd need to find a suitable equivalent set of operation (for example a set of any rotation), that overall are going to have the same effect.

I tried to look online for papers or reference but I was only able to find qubits to qudits mapping, not the other way around.

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  • $\begingroup$ Can you elaborate on your question and add more details? $\endgroup$
    – FDGod
    Commented Dec 7, 2023 at 8:52

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You are right that you can think of your state as a single qudit of dimension $d=4$ or alternatively as a register of two $d=2$. In your specific example what you call $X^{1,2}$ can also be thought of as a SWAP gate acting on the two qubits.

That is, the function mapping $$\frac{1}{\sqrt 2}|00\rangle+\frac{1}{2}|01\rangle+\frac{1}{2}|11\rangle$$

to $$\frac{1}{\sqrt 2}|00\rangle+\frac{1}{2}|10\rangle+\frac{1}{2}|11\rangle$$ also swaps the contents of the two qubits.

$$\mbox{SWAP}=\begin{pmatrix} 1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}.$$

You might also have to worry about endianness conventions.

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