Algorithm 8: Shor [17]

Proceed with the following steps to find a candidate for the quotient $s/r$ where s is randomand r is the period of $a^x$ mod N. Prepare a $2n$-qubit query register in the state $\left|0\right.〉$ and an n-qubit answer register in the state $\left|1\right.〉$. Apply the quantum Fourier transform to the query register. Apply the specific unitary $U_{a,N}\left|x\right.〉\left|y\right.〉=\left|x\right.〉\left|y\times a^x\mathrm{mod}N\right.〉$. Apply the inverse quantum Fourier transform to the query register. Measure the query register to retrieve a candidate for $s/r$.   Return that candidate for $s/r$.