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Quantum-Inspired Kernel Learning for Sparse Data

Jincheng Zhang

Published Sep 3, 2026• 0 citations
Publisher

Abstract

Kernel methods have demonstrated remarkable success in various machine learning tasks, including classification and regression. However, a significant limitation of traditional kernel methods lies in their computational complexity, particularly when dealing with sparse datasets. The need to construct and invert dense kernel matrices, denoted as $K$, where $K_{ij} = k(s_i, s_j)$ with kernel function $k$, leads to a quadratic scaling problem, rendering these methods impractical for datasets with a large number of features or data points. This paper introduces a novel kernel learning algorithm, termed Quantum-Inspired Kernel Learning (QIKL), which addresses this challenge by mimicking quantum interference. QIKL leverages the inherent correlations within sparse datasets to efficiently learn kernel functions without explicitly constructing dense kernel matrices. The algorithm employs a probabilistic approach, represented by a set of quantum-inspired amplitudes, $(a_1, a_2, ..., a_K)$, where $K \approx \sum_{i=1}^{K} a_i^2$, to approximate the kernel matrix. The core idea is to utilize the interference patterns generated by these amplitudes to capture the relationships between data points. This approach significantly reduces the computational cost and memory requirements, making kernel methods viable for sparse data scenarios. We demonstrate the effectiveness of QIKL through theoretical analysis and a simplified numerical example, highlighting its potential for scaling to large-scale sparse data problems.

DOI
10.5281/zenodo.22267594

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Graph kernels are essential for learning the properties of graph data, a fundamental task in the analysis of such data. However, a major limitation of current graph kernel methods is their high computational cost, which becomes prohibitive when dealing with large-scale graphs. In order to address this bottleneck, we propose a quantum algorithm designed to analyze the intrinsic structural information within graph data through an efficient quantum kernel method. The core idea of our kernel method involves encoding pertinent information from graphs into quantum states, utilizing quantum kernel functions to represent the similarities among various graphs. This process is facilitated through the application of quantum walks and quantum signal processing techniques. Building upon this framework, we propose a quantum kernel ridge regression algorithm that consists of two key steps. First, we employ our quantum kernel method to prepare a quantum state, in which the amplitudes encode the kernel functions. Second, we implement quantum feature mapping in parallel to encode the corresponding kernel matrix into the quantum circuit. Crucially, we demonstrate that the computational complexity of our regression algorithm scales primarily as O(M3/2)O({M}^{3/2})O(M3/2) with respect to the size of the graph set MMM. This suggests that our quantum kernel method could process graph data with resource requirements scaling approximately as O(polylog(N))O(\text{poly}log(N))O(polylog(N)), where NNN denotes the number of nodes in the graph. Accordingly, this underscores the advantage of our approach over classical methods in the context of large-scale graph analysis. To validate our approach, we conduct numerical experiments on chemical molecular datasets, demonstrating the method's effectiveness and advantages. In summary, this work represents a significant advancement in quantum graph learning, enhancing its potential for analyzing large-scale graph data.
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.
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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.