Structural Isomorphism Between LLM Embedding Spaces and Quantum Mechanical Systems

Abstract

Timo Aukusti Laine

Large language models (LLMs) represent semantic information as high-dimensional embedding vectors, typically compared using cosine similarity. This paper establishes a structural isomorphism between LLM embedding spaces and quantum mechanical systems. We demonstrate that the transformed cosine similarity can be precisely mapped to a quantum mechanical measurement probability, governed by a rank-1 Hamiltonian and the Schrödinger equation, where the time parameter is identified as a gauge variable. A commutative diagram unifies four equivalent expressions for this similarity, spanning classical bilinear forms and quantum Born rule probabilities. Crucially, the quantum formulation reveals a local U(1) gauge symmetry, absent from classical descriptions, with an associated conserved semantic charge modeling contextual influence. A quantum circuit on logarithmically many qubits yields this similarity as a single measurement probability, a result confirmed by simulation. The isomorphism is structural, not physical: it highlights that the mathematical structures governing LLM embeddings are precisely those of quantum mechanics, and this correspondence is exact.