A modular, adaptive, and scalable quantum factoring algorithm

Abstract

Shor’s algorithm for integer factorization offers an exponential speedup over classical methods, but remains impractical on noisy intermediate-scale quantum (NISQ) hardware due to the need for many coherent qubits and very deep circuits. Building on our recent work on adaptive and windowed phase estimation methods, we have developed a modular, windowed formulation of Shor’s algorithm that potentially mitigates these limitations by restructuring phase estimation into shallow, independent circuit blocks that can be executed sequentially or in parallel, followed by lightweight classical post-processing. This approach allows for a reduction in the size of the phase (or counting) register from a large number of qubits down to a small, fixed block size of only a few qubits (for example, three or four phase qubits were sufficient for the computational examples considered in this work), while leaving the work register requirement unchanged. The independence of the blocks allows for parallel execution and makes the approach more compatible with near-term hardware than the standard Shor’s formulation. An additional feature of the framework is the overlap mechanism, which introduces redundancy between blocks and enables robust reconstruction of phase information, though zero-overlap configurations can also succeed in certain regimes. Numerical simulations verify the correctness of the modular formulation while also showing substantial reductions in counting qubits per block. © The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026.