QuantumPhaseNet - Gauge-Covariant Geometric and Quantum-Spectral Semantic Concept Hierarchies - Abstract

Sep 15, 2026

Uploaded by Finn

QuantumPhaseNet presents a gauge-covariant geometric and quantum-spectral extension of Transformer representations for modeling context-dependent semantic states. Complex amplitudes represent semantic states, while a covariant phase rate yields a semantic wavelength as a proxy for conceptual scale and low-frequency graph modes define document-level discourse direction. The theory proves local gauge invariance, quantum-block unitarity, boundedness and conditional stability of WavePhase Attention, and a calibratable hallucination-risk formulation, validated offline in a synthetic Validation Studio.

LanguageEnglish

CategoryResearch & Report

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Abstract

Terminology and Scope of Claims

Main Contributions

Introduction

Background

Definitions of Realizations and Claims

arXiv :2608 . 15820v1 [ cs .CL] 16 Aug 2026 QuantumPhaseNet: A Gauge-Covariant Geometric and Quantum-Spectral Theory of Semantic Concept Hierarchies with Prototype Validation of a Classical Quantum-Inspired Model Kiyotaka Kasubuchi Kazuo Fukiya July 30, 2026 Abstract We present QuantumPhaseNet, a gauge-covariant geometric and quantum-spectral extension of Transformer representations. Context-dependent semantic states are modeled as complex amplitudes; a covariant phase rate induces a semantic wavelength used as a proxy for conceptual scale; and low-frequency graph modes define a document-level discourse direction. The theoretical part establishes local gauge invariance, unitarity of the quantum block, boundedness and conditional stability of WavePhase Attention, and a calibratable hallucination-risk formulation. We also implemented a fully offline Validation Studio for the classical quantum-inspired pipeline in Section 14.1 and evaluated the five research questions in Section 16.1 on its built-in synthetic setting (n=240, observation noise 0.22, circuit noise 0.08, five seeds) . RQ1 yielded a wavelength-hierarchy Spearman correlation of 0.852 versus 0.707 for the baseline, 87.3% direction accuracy, and AUC 0.953 . RQ2 achieved discourse alignment 0.933 versus 0.589 and 41.2 versus 16.2 paragraphs before drift. RQ3 achieved AUROC 0.881 versus cosine 0.765 and phase-shuffle 0.536 . RQ4 achieved error-detection AUROC 0.854 versus entropy 0.634, with Brier 0.150 and ECE 0.098 . RQ5 did not show quantum advantage: target probability and end-to-end cost eﬀiciency were 25.5% and 0. 107, compared with 70 .7% and 0 .707 for the Chebyshev classical approximation. These results provide initial synthetic evidence for the classical quantum-inspired components, but not external validity or unconditional quantum speedup. Keywords: Transformer; semantic manifold; gauge connection; covariant phase; semantic wavelength; concept hierarchy; graph Fourier transform; classical quantum-inspired computation; WavePhase Attention; hallucination; calibration; prototype validation 1 . Introduction 1.1 Background Transformers compute dependencies within a sequence in parallel through self-attention. For an input representation 􀔍 ∈ ℝ􀖉×􀕿 , standard self-attention is 􀔆 = 􀔍􀔌􀕲 , 􀔀 = 􀔍􀔌􀕬 , 􀔋 = 􀔍􀔌􀕷 , Att(􀔆, 􀔀, 􀔋 ) = softmax ( + 􀔂causal) 􀔋 as introduced by Vaswani et al. (2017) . Although this architecture is highly expressive, inner-product similarity alone does not explicitly reveal concept inclusion, levels of abstraction, parallel transport between contexts, document-level discourse direction, connection to evidence, or the geometry of a generation trajectory. Order Embeddings learn semantic hierarchies as explicit coordinate-wise orders (Vendrov et al. 2016), while Poincaré Embeddings exploit the correspondence between tree-like volume growth and negative curvature (Nickel and Kiela 2017) . Geometric deep learning incorporates symmetry and geometry into learning systems (Bronstein et al. 2021); graph Fourier analysis provides spectral tools on irregular domains (Shuman, Ricaud, and Vandergheynst 2013); and vector diffusion based on the connection Laplacian compares vectors that live in different tangent spaces (Singer and Wu 2012) . Quantum natural language processing likewise provides precedents for representing semantic composition with Hilbert spaces, category theory, tensor products, complex states, and quantum operators (Coecke, Sadrzadeh, and Clark 2010; Coecke et al. 2020) . Rather than merely juxtaposing these ideas, this paper integrates gauge-covariant phases of semantic states, a wavelength-based hierarchical scale, a low-frequency discourse direction, quantumspectral selection, attention, and hallucination risk into a single operator framework. 1.2 Terminology and Scope of Claims We call the overall model QuantumPhaseNet and its core attention mechanism WavePhase Attention. The present work extends our earlier

FAQs

What is QuantumPhaseNet designed to model?

QuantumPhaseNet models context-dependent semantic states and their hierarchical organization. It uses complex amplitudes, semantic wavelength, and low-frequency graph modes to capture conceptual scale and document-level discourse direction.

How does the paper extend Transformer representations?

It replaces a frame-dependent phase with a gauge-covariant phase, introduces a gauge connection and curvature on a complex semantic bundle, and adds spectral selection plus a calibratable hallucination-risk formulation.

Is the “quantum advantage” claimed as a general result?

No. The results provide conditional and synthetic evidence for classical quantum-inspired components, and the paper does not report unconditional quantum speedup in the evaluated setting.

Sep 15, 2026

Uploaded by Finn