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

TL;DR

This paper proposes a diffusion-based hybrid model for factoring, replacing quantum circuit evolution with hardware-like diffusion on a finite Cayley graph to perform order finding. The core contribution is a diffusion order-finding theorem: after $n_0=O((\log N)^2)$ diffusion steps, a single heat-kernel reading $p_{n_0}(e)$ determines the order $r=\operatorname{ord}_N(b)$ by rounding $1/p_{n_0}(e)$, enabling a Shor-style factoring route with polynomial digital work. It also develops a diffusion-assisted factoring algorithm, analyzes collision-based relation finding and gcd stabilization, and provides concrete numerical examples illustrating both diffusion-based and collision-based approaches. The results highlight how diffusion as a hardware primitive can emulate key spectral mixing properties to extract arithmetic structure, potentially offering a hardware-oriented alternative or complement to quantum approaches for order finding and factoring. An RC-network realization is discussed to map the diffusion primitive to a physical substrate, linking the abstract diffusion model to implementable hardware. Overall, the work clarifies the theoretical bounds, demonstrates practical feasibility on selected instances, and frames diffusion as a principled, locality-aware resource in factoring.

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.

Figures (7)

• Figure 1: Cayley graph on $\langle 3\rangle\subset(\mathbb{Z}/299\mathbb{Z})^\ast$ with generators $b^{\pm 2^t}$. Here $N=299$, $b=3$, and $\mathop{\mathrm{ord}}\nolimits_{299}(3)=33$.
• Figure 2: The identity value $p_n(e)$ for the half--lazy walk on $\langle 3\rangle$ (with $N=299$).
• Figure 3: The sequence $1/p_n(e)$ converging to $r=\mathop{\mathrm{ord}}\nolimits_{299}(3)=33$; rounding recovers $r$. (The dashed line indicates $r=33$.)
• Figure 4: A local view of the Cayley graph near the identity (shown in exponent coordinates). This is only for illustration; the full Cayley graph has $r=5313$ vertices.
• Figure 5: The identity heat-kernel value $p_n(e)$ as a function of the number $n$ of diffusion steps. As $n\to\infty$ one has $p_n(e)\to 1/r$.

Theorems & Definitions (38)

• Proposition 2.1
• proof
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• Lemma 3.4
• proof
• Lemma 3.5