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Suppose that a quantum state $|\psi\rangle$ in question is $|\psi\rangle \propto \sum_x |x\rangle|f(x)\rangle$.

I want to implement a quantum swap operation such that computational basis $|x\rangle|f(x)\rangle$ becomes $|x\rangle|f(x+1)\rangle$. That is, $|x\rangle|f(x)\rangle \to |x\rangle|f(x+1)\rangle$. This seems to be just about a series of basis swaps, but I am not so sure about this. Can we implement this as a quantum circuit? (For the greatest $x$, $x+1$ is considered to be zero.)

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Probably the best way to think about this is "subtract 1 from the first register". This is a standard classical computation, and hence you can write out a circuit for it. (Although you'll want to be careful of your boundary conditions.)

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