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.
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.
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.
A basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely, to efficiently perform computations in an intractably large Hilbert space. In this Letter we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyze the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualize the working principle with two-dimensional minibenchmark datasets.
This paper proposes a novel adaptive quantum kernel learning framework designed to overcome the limitations of classical kernel methods when applied to quantum state analysis. The core challenge lies in effectively mapping the high-dimensional space of quantum states into a manageable, low-dimensional feature space for traditional kernel-based learning algorithms. Existing methods often fail to capture the intricate correlations inherent in quantum states, leading to suboptimal results. Our approach introduces a self-adaptive kernel learning algorithm that dynamically adjusts the kernel function based on the observed quantum state correlations. This adaptation is achieved through a quantum-inspired optimization process utilizing variational quantum circuits. The circuits evolve the kernel parameters, guided by quantum state tomography data, effectively learning a kernel function tailored to the specific quantum system being analyzed. The key innovation is the integration of quantum mechanics directly into the kernel learning process, offering a potentially significant improvement in feature extraction and classification accuracy for quantum data. This work addresses a critical bottleneck in quantum machine learning, paving the way for more powerful and accurate quantum algorithms.
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.
Quantum machine learning (QML) models are aimed at learning from data encoded in quantum states.Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes.As such, it is fundamental to develop schemes that encode as much information as available about the problem at hand.In this work we present a simple, yet powerful, framework where the underlying invariances in the data are used to build QML models that, by construction, respect those symmetries.These so-called group-invariant models produce outputs that remain invariant under the action of any element of the symmetry group G associated with the dataset.We present theoretical results underpinning the design of G-invariant models, and exemplify their application through several paradigmatic QML classification tasks, including cases when G is a continuous Lie group and also when it is a discrete symmetry group.Notably, our framework allows us to recover, in an elegant way, several well-known algorithms for the literature, as well as to discover new ones.Taken together, we expect that our results will help pave the way towards a more geometric and group-theoretic approach to QML model design.
Quantum kernel methods are a promising branch of quantum machine learning, but their per- formance across data types and encoding choices remains incompletely understood. To address this gap, we developed an algorithmic framework for variational quantum kernels utilizing resource- efficient ansätze for complex classification tasks and introduced a parameter scaling technique to accelerate convergence. We conducted a comprehensive benchmark of this framework on eight chal- lenging, real-world and high-dimensional datasets covering tabular, image, time series, and graph data. Our results show that the proposed quantum kernel demonstrated competitive classification accuracy compared to standard classical kernels in classical simulation, such as the radial basis function (RBF) kernel. This work demonstrates that properly designed quantum kernels can func- tion as versatile, high-performance tools, laying a foundation for quantum-enhanced applications in machine learning. Further research is needed to fully assess the practical performance of quantum methods.