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

TL;DR

This paper provides a theoretical analysis showing that continuous chain-of-thoughts in transformers can solve graph reachability problems efficiently by encoding multiple search paths simultaneously, outperforming discrete methods.

Contribution

It proves that continuous CoTs enable parallel search in transformers, reducing steps needed for graph reasoning, and shows this superposition naturally emerges during training.

Findings

Continuous CoTs encode multiple search frontiers as superpositions.

Transformers with continuous CoTs solve graph reachability in D steps, where D is graph diameter.

Experimental results confirm the theoretical advantages and natural emergence of superpositions.

Abstract

Large Language Models (LLMs) have demonstrated remarkable performance in many applications, including challenging reasoning problems via chain-of-thoughts (CoTs) techniques that generate ``thinking tokens'' before answering the questions. While existing theoretical works demonstrate that CoTs with discrete tokens boost the capability of LLMs, recent work on continuous CoTs 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 CoTs 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 CoTs 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 CoTs must choose a single path sampled from the superposition state, which leads to sequential search that requires many more steps and may be trapped into 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 CoTs, without explicit supervision to guide the model to explore multiple paths simultaneously.