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I've been struggling to understand the modular exponent bit of Shor's algorithm. My understanding is that it takes a register in the state $\frac{1}{\sqrt{Q}}\sum_{k=1}^{Q-1} |k\rangle |0\rangle$ to the state $\frac{1}{Q}\sum_{k=1}^{Q-1} |k\rangle |f(k)\rangle$ where $f(k) = x^{k}$ mod $N$. (Here, $x$ is the random integer found at the start of Shor's algorithm.)

My question is: Why is this operation unitary?

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The critical thing, in this case, about a unitary operator is that it maps orthogonal states to orthogonal states (if $\langle i|j\rangle=0$, then $\langle i|U^\dagger U|j\rangle=0$, so the transformed vectors $U|i\rangle$ and $U|j\rangle$ are orthogonal). Now, you've defined what it must do for a set of states: $$ |k\rangle|0\rangle\mapsto |k\rangle|f(k)\rangle. $$ Now, it should be clear that all of the outputs are orthogonal to all other ones, simply as a result of the different values of $k$. So, surely, it will be possible to define a unitary over all possible basis states.

Indeed, the one of the usual starting points is to define the action of the unitary as $$ |k\rangle|y\rangle\mapsto \left\{\begin{array}{cc} |k\rangle|y+x^k\text{ mod }N\rangle & y<N \\ |k\rangle|y\rangle & y\geq N \end{array}\right. $$ (a second is to use $|k\rangle|yx^k\text{ mod }N\rangle$ on the top line). If we look at all possible $y$ and $k$, then we're looking at the whole basis. Again, different $k$s clearly give orthogonal vectors, independent of the value of $y$. Primarily, we have to check that $$ \langle y+x^k\text{ mod }N|y'+x^k\text{ mod}N\rangle=0. $$ This must be true provided $y-y'\text{ mod }N\neq0$, which is true for cases $y,y'<N$ and $y\neq y'$.

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