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If you have two registers in the state $\frac{1}{2^{n/2}} \sum_{x = 0}^{2^{n/2} - 1} |x\rangle |0\rangle$, how could you construct a gate that produces a superposition of states $|x\rangle|1\rangle$ when some integer $r$ divides $x$, and $|x\rangle|0\rangle$ otherwise, for each input?

I.e. a unitary quantum gate that replicates the function $f(x) = \begin{cases} 1 \text{ if } r \text{ divides } x\\ 0 \text{ otherwise} \end{cases}$

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  • $\begingroup$ Do you mean for the second to be $r|x$ cast to 0 or 1. X is being summed over, not a free variable. $\endgroup$ – AHusain Oct 20 '18 at 20:09
  • $\begingroup$ Yes, second register cast to a superposition of states $|0\rangle$ and $|1\rangle$, based on whether the first register is a multiple of $r$. First register is an equal superposition of all states. $\endgroup$ – nikojpapa Oct 20 '18 at 20:28
  • $\begingroup$ @AHusain Ah, I see what you mean. Does it make more sense after the edit? $\endgroup$ – nikojpapa Oct 20 '18 at 21:07
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    $\begingroup$ Also what is your gate set? Is $r$ arbitrary? $\endgroup$ – AHusain Oct 20 '18 at 21:12
  • $\begingroup$ Just the standard single-bit quantum gates, H, Paulis, $\pi/8$, etc. I'm using Q#. And yes, $r$ is arbitrary. $\endgroup$ – nikojpapa Oct 20 '18 at 21:15
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The simplest way would be to do a long division to compute the remainder, toggle the target bit if there is a non-zero remainder, then uncompute the long division.

Here is an example $r=3$, $N < 16$ circuit in Quirk:

mod 3 != 0 circuit

Note the displays on the right hand side, which show that conditioning on the output qubit (the bottom one) leaves only values divisible by 3 in the input register (the top 4 qubits).

The basic idea is to keep track of the maximum value $m$ that could possibly be in the input register $i$, then iteratively pick the largest $k$ such that $r^k \leq m$ and subtract $r^k$ out of $i$ if $i \geq r^k$. This reduces $m$ by $r^k$. Repeat this until $m < r$, then toggle your output bit if $i=0$. Then uncompute all the conditional subtractions to restore $i$.

A proper construction would not require $r$ subtractions for each value of $k$ as this one does, and a proper construction would expand the comparison and addition circuits into their full form, but I think this construction gets the right idea across.

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