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Quantum-Inspired Kernel Methods for Kernel Matrix Computation

Jincheng Zhang

Published Aug 31, 2026• 0 citations
Publisher

Abstract

Kernel matrix computation lies at the heart of numerous machine learning algorithms, particularly those employing kernel methods like Support Vector Machines (SVMs) and Gaussian Processes. However, the computational complexity of calculating these matrices, scaling quadratically with the number of data points (O(n²)), presents a significant bottleneck, especially for large datasets. This paper proposes a novel approach to kernel matrix computation inspired by quantum algorithms, specifically utilizing concepts from quantum annealing and adiabatic quantum computation. We introduce a heuristic algorithm that mimics the iterative process of quantum annealing to approximate the kernel matrix. The algorithm employs a belief propagation-like strategy, iteratively refining estimates of the kernel elements based on local interactions, aiming to converge towards the true kernel matrix. We demonstrate that this quantum-inspired method can achieve significant reductions in computational time compared to traditional methods, particularly for high-dimensional datasets. The theoretical analysis suggests a convergence rate dependent on the problem's structure and the annealing schedule. We provide a detailed mathematical framework for the algorithm and discuss its limitations and potential extensions. The key contribution is a practical and computationally efficient method for approximating kernel matrix computation, paving the way for the wider adoption of kernel methods in resource-constrained environments.

DOI
10.5281/zenodo.22184282

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This paper explores a novel approach to kernel design for machine learning, drawing inspiration from the principles of quantum entanglement. Traditional kernel functions in Support Vector Machines (SVMs) frequently necessitate extensive manual tuning and often fail to achieve optimal performance. We introduce a kernel design algorithm that mimics the correlated behavior observed in entangled quantum systems. The algorithm constructs kernel functions based on the correlated features of data points, effectively simulating entanglement effects to potentially enhance classification accuracy. This work presents a fundamental shift in kernel design, moving away from hand-crafted kernels towards a more adaptive and potentially more powerful approach guided by quantum-inspired concepts. The core contribution lies in the methodology for generating kernels that dynamically capture complex relationships within data, leading to improved classification outcomes. We demonstrate the potential of this approach through a theoretical framework and outline a design strategy that can be readily applied to various machine learning tasks.
This paper investigates the application of quantum-inspired optimization techniques to accelerate large-scale matrix computations. Traditional optimization algorithms often encounter significant challenges when dealing with the complexity inherent in these computations. We propose a novel approach leveraging concepts derived from quantum annealing and superposition to explore the solution space more efficiently. The core idea is to mimic the probabilistic exploration characteristic of quantum systems within a classical framework. Specifically, we utilize an iterative optimization process, adapting parameters based on simulated "quantum fluctuations," aiming to converge towards optimal solutions faster than conventional methods. The theoretical framework is presented, and we detail the implementation strategy, focusing on the design of a suitable objective function and a corresponding optimization algorithm. Preliminary results suggest the potential for significant speedups, particularly for computationally demanding matrix operations, though further investigation and refinement are needed. The key contribution lies in the direct translation of quantum-inspired principles into a practical optimization strategy for matrix computations.
Support vector machine (SVM) is a particularly powerful and flexible supervised learning model that analyzes data for both classification and regression, whose usual algorithm complexity scales polynomially with the dimension of data space and the number of data points. To tackle the big data challenge, a quantum SVM algorithm was proposed, which is claimed to achieve exponential speedup for least squares SVM (LS-SVM). Here, inspired by the quantum SVM algorithm, we present a quantum-inspired classical algorithm for LS-SVM. In our approach, an improved fast sampling technique, namely indirect sampling, is proposed for sampling the kernel matrix and classifying. We first consider the LS-SVM with a linear kernel, and then discuss the generalization of our method to nonlinear kernels. Theoretical analysis shows our algorithm can make classification with arbitrary success probability in logarithmic runtime of both the dimension of data space and the number of data points for low rank, low condition number, and high dimensional data matrix, matching the runtime of the quantum SVM.
Abstract Quantum machine learning, an emerging field at the intersection of quantum computing and classical machine learning, has shown great promise in enhancing computational capabilities beyond classical bounds. A key element in this area of research is the utilization of Quantum Kernel Estimators, traditionally grounded in the symmetries of SU(2) groups associated with qubits. This study extends the conceptual framework of Quantum Kernel Estimators to incorporate a broader spectrum of symmetry groups. By harnessing the various structures of Lie groups, we develop novel quantum-inspired feature maps that offer more flexible and potentially powerful ways to encode and compress classical data into quantum states. We present a comprehensive theoretical introduction for this approach, followed by a methodology that integrates the developed feature maps into kernel-based quantum classifiers. Our results, derived from a series of computational experiments across various datasets, demonstrate the efficacy of this approach in comparison to traditional quantum and classical machine learning models. The findings not only underline the versatility of Lie-group theory in enhancing quantum machine learning algorithms but also open new avenues for exploring complex symmetries in quantum information processing. This research bridges a gap between the study of symmetries and machine learning, paving the way for more sophisticated quantum algorithms capable of tackling complex, high-dimensional data in ways previously unattainable.
Support Vector Machines (SVMs) are powerful machine learning models widely used for classification and regression tasks. However, their performance can be significantly impacted by the choice of the kernel function, which defines how data points are mapped into a higher-dimensional space. Traditional kernels often struggle to capture complex, non-linear relationships effectively, leading to suboptimal results. This paper proposes a novel approach to kernel design inspired by quantum mechanical principles. Specifically, we leverage concepts of quantum entanglement and superposition to generate kernels that can potentially capture richer and more nuanced data patterns. Our core claim is that employing these quantum-inspired designs can improve SVM performance and scalability. We present a theoretical framework outlining how quantum mechanical effects can be translated into kernel functions. The proposed method is based on constructing kernel functions that implicitly encode correlations between data points, mimicking the inherent interconnectedness observed in quantum systems. We explore specific kernel formulations based on these principles and discuss their potential advantages and challenges. The ultimate goal is to demonstrate that quantum inspiration can unlock new avenues for kernel design, leading to more effective SVM models.