Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring

• URL: http://arxiv.org/abs/2601.02518v1
• Date: Mon, 05 Jan 2026 19:45:38 GMT
• Title: Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring
• Authors: Carlos A. Cadavid, Paulina Hoyos, Jay Jorgenson, Lejla Smajlović, J. D. Vélez,
• Abstract summary: We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an emphiterated diffusion process on a finite graph.<n>Our comparison with Shor's algorithm is emphconceptual and model-based.<n>We report complexity in two cost measures: digital steps and diffusion steps.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an \emph{iterated diffusion process} on a finite graph. Concretely, a \emph{diffusion step} is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an $\ell^{1}$--normalized state vector, followed by an optional readout of selected coordinates. Let $N\ge 3$ be an odd integer which is neither prime nor a prime power, and let $b\in(\mathbb{Z}/N\mathbb{Z})^\ast$ have odd multiplicative order $r={\rm ord}_N(b)$. We construct, without knowing $r$ in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup $\langle b\rangle$ and whose edges correspond to the powers $b^{\pm 2^t}$ for $t\le \lfloor \log_2 N\rfloor+1$. Using an explicit spectral decomposition together with an elementary doubling lemma, we show that $r$ can be recovered from a single heat-kernel value after at most $O((\log_2 N)^2)$ diffusion steps, with an effective bound. We then combine this order-finding model with the standard reduction from factoring to order finding (in the spirit of Shor's framework) to obtain a randomized factorization procedure whose success probability depends only on the number $m$ of distinct prime factors of $N$. Our comparison with Shor's algorithm is \emph{conceptual and model-based}. We replace unitary $\ell^2$ evolution by Markovian $\ell^1$ evolution, and we report complexity in two cost measures: digital steps and diffusion steps. Finally, we include illustrative examples and discussion of practical implementations.

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