Frequency-Domain Analysis of Time-Dependent Multiomic Data in Progressive Neurodegenerative Diseases: A Proposed Quantum-Classical Hybrid Approach with Quaternionic Extensions

Abstract

Progressive neurodegenerative diseases, including Alzheimer's disease (AD), multiple sclerosis (MS), Parkinson's disease (PD), and amyotrophic lateral sclerosis (ALS), exhibit complex, nonlinear trajectories that challenge deterministic modeling. Traditional time-domain analyses of multiomic and neuroimaging data often fail to capture hidden oscillatory patterns, limiting predictive accuracy. We propose a theoretical mathematical framework that transforms time-series data into frequency or s-domain using Fourier and Laplace transforms, models neuronal dynamics via Hamiltonian formulations, and employs quantum-classical hybrid computing with variational quantum eigensolvers (VQE) for enhanced pattern detection. This theoretical construct serves as a foundation for future empirical works in quantum-enhanced analysis of neurodegenerative diseases. We extend this to quaternionic representations with three imaginary axes ($i, j, k$) to model multistate Hamiltonians in multifaceted disorders, drawing from quantum neuromorphic computing to capture entangled neural dynamics \citep{Pehle2020, Emani2019}. This approach leverages quantum advantages in handling high-dimensional amplitude-phase data, enabling outlier detection and frequency signature analysis. Potential clinical applications include identifying high-risk patients with rapid progression or therapy resistance using s-domain biomarkers, supported by quantum machine learning (QML) precedents achieving up to 99.89% accuracy in Alzheimer's classification \citep{Belay2024, Bhowmik2025}. This framework aims to lay the groundwork for redefining precision medicine for neurodegenerative diseases through future validations.