Integer Factoring with Unoperations

TL;DR

This work introduces the unoperation framework, defining how to recover all inputs that map to a given output for a given operation, and applies it to quantum circuits via unaddition and unmultiplication. By constructing a quantum unadder (with a fixed qubit budget of $O((\log N)^2)$) and an optimized $3$-qubit gate, the authors demonstrate a ripple-carry approach that reverses the flow of information and yields a superposition over all valid input decompositions. The unmultiplication circuit extends this idea to factorization, enabling a quantum device to produce all factor pairs for a product $p$ with the same $qubit scaling, supported by simulations up to sizable bit-lengths. Overall, the paper provides a constructive, simulated pathway to quantum factoring based on unoperations, highlighting potential practical impact and guiding future exploration of trapdoor-function reversibility in quantum computing.

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.

Figures (5)

• Figure 1: Black box circuit of a full-unadder. It is the unoperation device corresponding to the full-adder used in classical circuit design construction for . It is capable of the unaddition of a 1 sum with the carry $c_{\text{out}}$, which will produce valid bit combinations for $a$, $b$, and $c_{\text{in}}$ satisfying Eq. \ref{['eq:Full-Adder']}.
• Figure 2: A quantum circuit of a full-unadder. The action on $\ket{c_{\text{out}}}\ket{sum} = \ket{00}$ is implicit and the actions on $\ket{c_{\text{out}}}\ket{sum} \in \{ \ket{01}, \ket{10}, \ket{11} \}$ (cf. Tab. \ref{['tab:Full-UnadderTruthTable']}) are described by the sections of the circuit indicated by the separation lines, respectively.
• Figure 3: Black box circuit of a 3 unmultiplier. It is the unoperation device corresponding to the classical circuit for the multiplier built from and in turn full-adders (cf.IntroLogicCircuits-ArithmeticCircuits). It is capable of the unmultiplication of numbers whose factors fit into the 3 registers. The "enable" describes the binary multiplication with $1$ or $0$, i.e. if the enable signal is $0$ ($1$) the corresponding is off (on).
• Figure 4: Quantum circuit of a 3 with input sum $6$ (binary $110$). It uses the optimised full-unadder gate $\mathfrak{Un}+_{\text{opt}}$ implemented with the unitary matrix from Eq. \ref{['eq:OptFull-UnadderMatrix']}.
• Figure 5: Quantum circuit of a 3 unmultiplier with input product $6$ (binary $000110$, leading zeroes implicit in the circuit for clarity) with to recover factor $y$. It uses 3 denoted $\mathfrak{Un}+_{\text{opt},3\:\mathrm{bit}_i}$ for $i \in \{0,1,2\}$, which are implemented with optimised full-unadder gate $\mathfrak{Un}+_{\text{opt}}$ analogously to Fig. \ref{['fig:RCU6QCircuit']}.