arXiv subjects

Timo Aukusti Laine

Publications and source records attributed to Timo Aukusti Laine.

8 recordsLinked to original sources

arXiv2025

Discrete Semantic States and Hamiltonian Dynamics in LLM Embedding Spaces

We investigate the structure of Large Language Model (LLM) embedding spaces using mathematical concepts, particularly linear algebra and the Hamiltonian formalism, drawing inspiration from analogies with quantum mechanical systems. Motivated by the observation that LLM embeddings exhibit distinct states, suggesting discrete semantic representations, we explore the application of these mathematical tools to analyze semantic relationships. We demonstrate that the L2 normalization constraint, a characteristic of many LLM architectures, results in a structured embedding space suitable for analysis using a Hamiltonian formalism. We derive relationships between cosine similarity and perturbations of embedding vectors, and explore direct and indirect semantic transitions. Furthermore, we explore a quantum-inspired perspective, deriving an analogue of zero-point energy and discussing potential connections to Koopman-von Neumann mechanics. While the interpretation warrants careful consideration, our results suggest that this approach offers a promising avenue for gaining deeper insights into LLMs and potentially informing new methods for mitigating hallucinations.

cs.LG↗

arXiv2025

Quantum LLMs Using Quantum Computing to Analyze and Process Semantic Information

We present a quantum computing approach to analyzing Large Language Model (LLM) embeddings, leveraging complex-valued representations and modeling semantic relationships using quantum mechanical principles. By establishing a direct mapping between LLM semantic spaces and quantum circuits, we demonstrate the feasibility of estimating semantic similarity using quantum hardware. One of the key results is the experimental calculation of cosine similarity between Google Sentence Transformer embeddings using a real quantum computer, providing a tangible demonstration of a quantum approach to semantic analysis. This work reveals a connection between LLMs and quantum mechanics, suggesting that these principles can offer new perspectives on semantic representation and processing, and paving the way for future development of quantum algorithms for natural language processing.

quant-ph↗

arXiv2025

The Quantum LLM: Modeling Semantic Spaces with Quantum Principles

In the previous article, we presented a quantum-inspired framework for modeling semantic representation and processing in Large Language Models (LLMs), drawing upon mathematical tools and conceptual analogies from quantum mechanics to offer a new perspective on these complex systems. In this paper, we clarify the core assumptions of this model, providing a detailed exposition of six key principles that govern semantic representation, interaction, and dynamics within LLMs. The goal is to justify that a quantum-inspired framework is a valid approach to studying semantic spaces. This framework offers valuable insights into their information processing and response generation, and we further discuss the potential of leveraging quantum computing to develop significantly more powerful and efficient LLMs based on these principles.

cs.AI↗

arXiv2025

Semantic Wave Functions: Exploring Meaning in Large Language Models Through Quantum Formalism

Large Language Models (LLMs) encode semantic relationships in high-dimensional vector embeddings. This paper explores the analogy between LLM embedding spaces and quantum mechanics, positing that LLMs operate within a quantized semantic space where words and phrases behave as quantum states. To capture nuanced semantic interference effects, we extend the standard real-valued embedding space to the complex domain, drawing parallels to the double-slit experiment. We introduce a "semantic wave function" to formalize this quantum-derived representation and utilize potential landscapes, such as the double-well potential, to model semantic ambiguity. Furthermore, we propose a complex-valued similarity measure that incorporates both magnitude and phase information, enabling a more sensitive comparison of semantic representations. We develop a path integral formalism, based on a nonlinear Schrödinger equation with a gauge field and Mexican hat potential, to model the dynamic evolution of LLM behavior. This interdisciplinary approach offers a new theoretical framework for understanding and potentially manipulating LLMs, with the goal of advancing both artificial and natural language understanding.

cs.CL↗

arXiv2023

Anomaly and Brownian fluid particle in Navier-Stokes turbulence

We investigate the Navier-Stokes turbulence driven by a stochastic random Gaussian force. Using a field-theoretic approach, we uncover an anomaly that brings hidden structure to the theory. The anomaly is generated by a non-self-adjoint operator of the Jacobian and it follows the symmetries of the stochastic Navier-Stokes equation. We calculate the anomaly and demonstrate that by forcing the anomaly to vanish, the velocity field is constrained and a monopole-type object with a constant charge is formed. When the viscosity is zero, the anomaly can be interpreted as the Brownian damping coefficient of a random fluid particle. We provide the Brownian particle equation and its solution in the presence of a pump and viscosity. Our results suggest that the anomaly is an inherent feature of stochastic turbulence and must be taken into account in all stochastic turbulence calculations. This constitutes an additional law for the original set of stochastic Navier-Stokes equations.

physics.flu-dyn↗

arXiv2023

Local symmetries, anomalies and constrains in Burgers Turbulence

We study stochastic Burgers turbulence without pressure. We first show that the variational derivative of the Burgers equation is dependent on the velocity field, suggesting the existence of an anomaly. The anomaly is created by an operator that is non-self-adjoint. To calculate it correctly, we need to find its square. There are similarities with conformal and chiral two-dimensional field theories, but causality is the key that makes the difference. We calculate the determinant and use two local symmetries to verify the result. By requiring the disappearance of the anomaly, the velocity field is constrained and the theory becomes anomaly-free. These symmetries obey Kolmogorov's second law of self-similarity. One can choose an anomaly-free theory, a partially broken theory, or a fully broken theory by choosing the constraint appropriately. There is an analogy to gauge fixing or vacuum selection which define the local configuration.

hep-th↗

arXiv2020

Exact solutions and magnetic monopoles in adiabatic three level systems

We investigate the geometrical phase associated to the Schrödinger equation in a three level system in Stimulated Raman Adiabatic Passage (STIRAP). We solve explicitly a dual model, in which the pulses are applied in the counterintuitive and intuitive order. We show that when the pulse areas are finite, a pair of magnetic monopoles with opposite charges are created resulting the oscillations of the populations on the final states. The applications of the phase shift include, for example, phase gates in quantum computing, phase manipulation in quantum cryptography and phase interactions in quantum interference.

quant-ph↗

arXiv2013

Symmetries of the Burgers Turbulence without Pressure

We investigate local symmetries of the Burgers turbulence driven by an external random force. By using a path integral formalism, we show that the Jacobian has physics in it; local symmetries and an anomaly. We also study a local invariance of the effective action and show it is related to Kolmogorov's second law of self-similarity.

hep-th↗