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.

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

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.