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Suppose you have a qudit with $n$ decoherent states.

How would you name the gate which maps $|x\rangle\rightarrow |(x+1)\text{mod}\hspace{1mm}n\rangle$

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    $\begingroup$ What do you mean by decoherent states? As to your question what's wrong with calling the gate a $+1$ gate? $\endgroup$ Mar 8 at 19:01

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I am assuming you meant to say qudit with $n$-basis states. The gate you are describing would be a shift matrix, i.e., a generalized Pauli $X$.

As I have answered in your previous question (and also here), this would be a $X(1)$ gate.

$$X(1)|j\rangle = |(j +1) \text{mod} (n) \rangle \,,$$

where $\{|j\rangle\}$ are the basis states. (using different variable than one in your question to avoid confusion)

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