Can PCE solve the factorisation problem via optimisation?

Abstract

The ongoing progress in quantum technologies has fueled a sustained exploration of their potential applications across various domains, particularly in computational problems that are considered intractable for classical systems. Among these problems, integer factorisation remains of special interest due to its relevance to widely used cryptographic schemes such as RSA. Among the different possibilities, one approach to factorisation is to convert the problem into a binary optimisation problem. However, current proposals usually need a large number of qubits that make them unfeasible within the current hardware. In this work, we investigate a possible adaptation of the Pauli Correlation Encoding (PCE) algorithm to the factorisation problem. Due to its compression capability, it can drastically reduce the number of needed qubits. The proposed approach explores how the structure and dynamics of the PCE framework may be employed to encode and analyze candidate factor relations within a quantum computational setting. Rather than presenting a replacement for established quantum factorisation methods, this study aims to provide a preliminary examination of the feasibility and limitations of the proposed adaptation. We discuss the algorithmic design, its conceptual relationship with existing quantum approaches, and the practical constraints associated with implementation on current or near-term quantum hardware. Initial observations suggest that the method may offer an alternative perspective for studying factorisation within the broader context of quantum computation, although no claim is made regarding computational advantage. These results are intended primarily as an exploratory contribution to ongoing research in quantum algorithms and computational number theory.