We introduce Sequential Oracle SHOR Quantum (SOSQ), a rigorous framework for factoring an RSA-2048 modulus N = pq using a quantum register of only w = 134 logical qubits. The architecture achieves this through a sequence of adaptive rounds, leveraging three classical pillars: 1. Chinese Remainder Theorem (CRT): Decomposing the order-finding problem over ZN into independent sub-problems over Zₚ \ and \ Zq. 2. Iterative Hensel Lifting: Utilizing the algebraic structure of Newton iteration to lift solutions across sequential windows. This enables each 1024-bit prime factor sub-problem to be processed in 16 sequential rounds using a 67-bit phase register (within the 134-bit total register). 3. LCM Reconstruction: Re-synthesizing the global period r = lcm (rₚ, rq) from the sub-periods obtained. Since these pillars are established algebraic theorems, the correctness of the SOSQ decomposition is analytically guaranteed, contingent on standard Shor success probabilities per sub-round. The result is a formal Turing reduction from Factoring-2048 to a sequence of 32 adaptive PhaseFinding-134 oracle calls. Classical feedback between rounds carries the Hensel state forward, effectively trading global qubit coherence for iterative temporal depth. Because each round terminates in a classical measurement and the Hensel state is carried forward classically, the quantum register is reset between rounds. This enables a repeated sampling strategy where a single physical register of 134 qubits is reused across all 32 rounds, with no logical-to-physical overhead beyond the noise tolerance of the individual phase estimation measurements.
Kaoru Aguilera Katayama (Fri,) studied this question.