🤖 AI Summary
This work addresses the integer factorization problem by introducing a novel “unoperation” paradigm grounded in quantum reversible computation. Methodologically, it constructs a quantum circuit for the inverse of addition (unaddition) and leverages it to design the first dedicated quantum unmultiplier, thereby recasting factorization as an input-inversion problem over a reversible mapping. The key contribution is the systematic introduction of a reversible-operation framework: the unmultiplier coherently maps a given product state back to a superposition of all possible factor pairs, enabling parallel inversion-based factor search. Crucially, the architecture requires only O((log N)²) qubits—achieving state-of-the-art resource efficiency—and offers a conceptually distinct, hardware-efficient alternative to Shor’s algorithm for quantum integer factorization.
📝 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.