Integer Factoring with Unoperations

Abstract

This work introduces the notion of unoperation $\mathfrak{Un}(\hat{O})$ of some operation $\hat{O}$. Given a valid output of $\hat{O}$, the corresponding unoperation produces a set of all valid inputs to $\hat{O}$ that produce the given output. Further, the working principle of unoperations is illustrated using the example of addition. A device providing that functionality is constructed utilising a quantum circuit performing the unoperation of addition - referred to as unaddition. To highlight the potential of the approach the unaddition quantum circuit is employed to construct a device for factoring integer numbers $N$, which is then called unmultiplier. This approach requires only a number of qubits $\in \mathcal{O}((\log N{)}^2)$, rivalling the best known factoring algorithms to date.