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I am trying to implement Simon's algorithm which calls for a 2-to-1 mapping function that satisfies $f(x) = f(x⊕s)$.

I am looking for a simple way to code the oracle (using $H$, $Cx$, and $R$ gates), ideally with an easy way to redefine $s$.

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You could do something like:

  • assume the most significant bit of $s$ is 1.

  • write a function that says "if the most significant bit of $x$ is 0, return $x$. if the most significant bit of $x$ is 1, return $x\oplus s$.

This is easily implemented because you start by doing a transversal set of cNOT gates to copy $x$ from the input register to the output register. Then, you simply do a bunch of controlled-not gates controlled off the most significant bit of the first register, targeting each of the qubits in the output register for which the corresponding bit of $s$ is 1.

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  • $\begingroup$ how is the most significant bit chosen/defined? $\endgroup$ – Ronen Raz Jun 25 at 11:55
  • $\begingroup$ Well, really, you can make it any bit for which $s$ is 1. $\endgroup$ – DaftWullie Jun 25 at 15:06

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