Reasoning by Superposition: A Theoretical Perspective on Chain of Continuous Thought

Large Language Models (LLMs) have demonstrated remarkable performance in many applications, including challenging reasoning problems via chain-ofthought (CoT) techniques that generate "thinking tokens" before answering the questions. While existing theoretical works demonstrate that CoT with discrete tokens boosts the capability of LLMs, recent work on continuous CoT lacks a theoretical understanding of why it outperforms discrete counterparts in various reasoning tasks, such as directed graph reachability, a fundamental graph reasoning problem that includes many practical domain applications as special cases. In this paper, we prove that a two-layer transformer with D steps of continuous CoT can solve the directed graph reachability problem, where D is the diameter of the graph, while the best known result of constant-depth transformers with discrete CoT requires O(n 2 ) decoding steps where n is the number of vertices (D < n). In our construction, each continuous thought vector is a superposition state that encodes multiple search frontiers simultaneously (i.e., parallel breadth-first search (BFS)), while discrete CoT must choose a single path sampled from the superposition state, which leads to a sequential search that requires many more steps and may be trapped in local solutions. We also performed extensive experiments to verify that our theoretical construction aligns well with the empirical solution obtained via training dynamics. Notably, encoding of multiple search frontiers as a superposition state automatically emerges in training continuous CoT, without explicit supervision to guide the model to explore multiple paths simultaneously. Our code is available at https://github.com/Ber666/reasoning-by-superposition.

Large language models (LLMs) have shown strong performance in many reasoning tasks, especially when empowered with chain-of-thought (CoT) [Wei et al., 2022] (e.g., hard problems like AIME and math proving). However, they also struggle with tasks that require more sophisticated reasoning capability [