Quantum Circuit Simulation of Compartmental Drug Dynamics: Leveraging Variational Algorithms for Nonlinear Mixed-Effects Population Pharmacokinetics

Quantitative Structure-Activity Relationship ($\mathtt{QSAR}$) modeling is a foundational computational methodology in early-stage drug discovery, heavily relied upon for predicting compound toxicity, bioavailability, and therapeutic potential. However, classical methods often struggle to effectively map the highly complex, non-linear, and high-dimensional interactions inherent in molecular data, leading to reduced predictive accuracy and costly late-stage clinical failures. In this paper, we present a Quantum Multiple Kernel Learning ($\mathtt{QMKL}$) framework, dubbed Next-Gen $\mathtt{Q^2SAR}$, that leverages Quantum Support Vector Machines ($\mathtt{QSVMs}$) to overcome these classical limitations. By encoding molecular descriptors into exponentially large quantum Hilbert spaces, our approach substantially enhances the expressiveness of non-linear modeling. Benchmarking our quantum-enhanced framework on a dataset targeting the $\mathtt{DYRK1A}$ kinase (a critical target for Alzheimer's disease), the $\mathtt{QMKL}$-$\mathtt{SVM}$ achieves an impressive Area Under the Curve ($\mathtt{AUC}$) score of $0.8750$, significantly outperforming classical state-of-the-art Gradient Boosting models ($\mathtt{AUC} = 0.8037$). Furthermore, we establish a theoretical and empirical pathway toward resolving classical data bottlenecks through projected quantum kernels ($\mathtt{PQK}$) and measurement accelerators. As quantum computing architecture matures, this framework paves the way for autonomous cognitive architectures and self-improving drug discovery pipelines, promising to unlock deeper insights across vast chemical spaces and to accelerate the development of life-saving therapeutics.

We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.

Many applications in quantum simulation, quantum chemistry, and quantum machine learning require not a single quantum state but an ensemble of states characterizing the heterogeneity of a target system. Preparing such ensembles state-by-state is prohibitive in both variational and fault-tolerant settings, thereby motivating a generative modeling approach. We introduce latent-conditioned parameterized quantum circuits (LPQCs), a hybrid quantum-classical framework in which classical neural networks map a latent variable sampled from a prior distribution to the parameters of a parameterized quantum circuit. We prove that LPQCs are universal approximators for probability measures over density operators in the 1-Wasserstein distance, extending classical universal approximation theorems to the quantum-distribution setting. We additionally introduce a multimodal latent prior and a mixture-of-experts circuit architecture, and show empirically that the latent-conditioned parameterization alleviates the barren plateau problem during optimization, a behavior for which we provide rigorous partial guarantees. Numerical experiments validate the framework on a synthetic multi-cluster ensemble of mixed quantum states and on a QM9-derived ensemble of 3-D molecular structures. In these tasks, LPQC outperforms recent quantum generative baselines and matches the generation quality of a classical neural-network baseline, while requiring an output dimension that grows only linearly with the number of qubits rather than exponentially. By leveraging classical expressivity in the latent space, LPQCs offer a tractable route to quantum generative modeling.