We investigate the runtime of quantum kernel estimation in the view of quantum kernel concentration effect. The study is performed for projected quantum kernel family evaluated on hyperspectral remote sensing data. The effect of exponential value concentration leads to the indistinguishability of the kernel matrix entries as the size of the quantum device grows. In order to prevent that, kernel values have to be estimated with a better precision. Increasing precision inevitably connects to an increasing number of circuit runs, which influences the runtime of quantum algorithm. This, in turn, frequently obstructs a possible advantage for quantum machine learning methods. We find that, against popular opinions, the effect of exponential value concentration does not rule out the utility of quantum kernel methods and the severity of the issue depends highly on the data used.
We investigate the runtime of quantum kernel estimation in the view of quantum kernel concentration effect. The study is performed for projected quantum kernel family evaluated on hyperspectral remote sensing data. The effect of exponential value concentration leads to the indistinguishability of the kernel matrix entries as the size of the quantum device grows. In order to prevent that, kernel values have to be estimated with a better precision. Increasing precision inevitably connects to an increasing number of circuit runs, which influences the runtime of quantum algorithm. This, in turn, frequently obstructs a possible advantage for quantum machine learning methods. We find that, against popular opinions, the effect of exponential value concentration does not rule out the utility of quantum kernel methods and the severity of the issue depends highly on the data used.
Quantum Machine Learning (QML) has gathered significant attention through approaches like Quantum Kernel Machines. While these methods hold considerable promise, their quantum nature presents inherent challenges. One major challenge is the limited resolution of estimated kernel values caused by the finite number of circuit runs performed on a quantum device. In this study, we propose a comprehensive system of rules and heuristics for estimating the required number of circuit runs in quantum kernel methods. We introduce two critical effects that necessitate an increased measurement precision through additional circuit runs: the spread effect and the concentration effect. The effects are analyzed in the context of fidelity and projected quantum kernels. To address these phenomena, we develop an approach for estimating desired precision of kernel values, which, in turn, is translated into the number of circuit runs. Our methodology is validated through extensive numerical simulations, focusing on the problem of exponential value concentration. We stress that quantum kernel methods should not only be considered from the machine learning performance perspective, but also from the context of the resource consumption. The results provide insights into the possible benefits of quantum kernel methods, offering a guidance for their application in quantum machine learning tasks.
The industry of quantum technologies is rapidly expanding, offering promising opportunities for various scientific domains. Among these emerging technologies, Quantum Machine Learning (QML) has attracted considerable attention due to its potential to revolutionize data processing and analysis. In this paper, we investigate the application of QML in the field of remote sensing. It is believed that QML can provide valuable insights for analysis of data from space. We delve into the common beliefs surrounding the quantum advantage in QML for remote sensing and highlight the open challenges that need to be addressed. To shed light on the challenges, we conduct a study focused on the problem of kernel value concentration, a phenomenon that adversely affects the runtime of quantum computers. Our findings indicate that while this issue negatively impacts quantum computer performance, it does not entirely negate the potential quantum advantage in QML for remote sensing.
Kernel methods in Quantum Machine Learning (QML) have recently gained significant attention as a potential candidate for achieving a quantum advantage in data analysis. Among other attractive properties, when training a kernel-based model one is guaranteed to find the optimal model's parameters due to the convexity of the training landscape. However, this is based on the assumption that the quantum kernel can be efficiently obtained from quantum hardware. In this work we study the performance of quantum kernel models from the perspective of the resources needed to accurately estimate kernel values. We show that, under certain conditions, values of quantum kernels over different input data can be exponentially concentrated (in the number of qubits) towards some fixed value. Thus on training with a polynomial number of measurements, one ends up with a trivial model where the predictions on unseen inputs are independent of the input data. We identify four sources that can lead to concentration including expressivity of data embedding, global measurements, entanglement and noise. For each source, an associated concentration bound of quantum kernels is analytically derived. Lastly, we show that when dealing with classical data, training a parametrized data embedding with a kernel alignment method is also susceptible to exponential concentration. Our results are verified through numerical simulations for several QML tasks. Altogether, we provide guidelines indicating that certain features should be avoided to ensure the efficient evaluation of quantum kernels and so the performance of quantum kernel methods.
Quantum kernel methods have emerged as a promising approach for leveraging high-dimensional feature spaces in machine learning, particularly in domains where classical kernel methods face scalability limitations. In this work, we present the first large-scale study of fidelity-quantum-kernel support vector machines for hyperspectral data classification without requiring heavy prior feature selection or dimensionality reduction. By simulating quantum kernels using tensor network contraction techniques and GPU acceleration, we overcome the computational bottlenecks traditionally associated with quantum models, achieving quadratic scaling O(n^2) in the number of qubits. Our approach enables the evaluation of quantum kernels on hyperspectral data with hundreds of spectral bands, aligning quantum feature spaces with real-world remote sensing applications. We provide an in-depth analysis of kernel bandwidth optimization, demonstrating its crucial role in mitigating exponential concentration effects and ensuring the model's ability to generalize. Experimental results on binary classification (Indian Pines and Methane Detection) and multiclass classification (Indian Pines) demonstrate that quantum kernels achieve competitive performance compared to a broad range of state-of-the-art classical baselines. As illustrative cases, on four 50-band splits selected from Indian Pines, the quantum model achieved a 78.0 pm6.2% accuracy for a binary classification task compared to 72.0 pm5.0% for the standard radial basis function (RBF) kernel. For a four-class classification task, the quantum kernel reached 83.3 pm3.1% accuracy, outperforming several state-of-the-art baselines. On five 75-band splits selected from the Methane Detection dataset, the quantum approach yielded 58.5\pm5.0% accuracy versus 55.1\pm2.5% for the classical counterpart...
Quantum support vector machines are classification algorithms that rely on quantum-generated kernels. The fidelity quantum kernel commonly used in quantum support vector machines suffers from exponential concentration as system size increases, preventing an efficient scaling beyond fewqubit systems. We introduce the Hamming quantum kernel, a classical post-processing method that is based on the same measurement outcomes as the fidelity quantum kernel. However, it avoids the exponential concentration problem by using the full measurement statistics rather than a single fidelity value. We evaluate the approach on both classical data (MNIST) and synthetic data generated from quantum circuits, using systems ranging from 2 to 27 qubits. Throughout the simulations, the Hamming quantum kernel outperforms the fidelity quantum kernel whenever 15 or more qubits are used. Furthermore, for synthetic quantum data, our method consistently outperforms the classical Gaussian kernel. This demonstrates that the Hamming quantum kernel improves the expressivity and robustness at larger qubit scales without requiring any additional quantum ressources.
Quantum kernel methods have emerged as a promising approach for leveraging high-dimensional feature spaces in machine learning, particularly in domains where classical kernel methods face scalability limitations. In this work, we present the first large-scale study of fidelity-quantum-kernel support vector machines for hyperspectral data classification without requiring heavy prior feature selection or dimensionality reduction. By simulating quantum kernels using tensor network contraction techniques and graphics processing unit acceleration, we overcome the computational bottlenecks traditionally associated with quantum models, achieving quadratic scaling O(n2) in the number of qubits. Our approach enables the evaluation of quantum kernels on hyperspectral data with hundreds of spectral bands, aligning quantum feature spaces with real-world remote sensing applications. We provide an in-depth analysis of kernel bandwidth optimization, demonstrating its crucial role in mitigating exponential concentration effects and ensuring the model's ability to generalize. Experimental results on binary classification (Indian Pines and methane detection) and multiclass classification (Indian Pines) demonstrate that quantum kernels achieve competitive performance compared to a broad range of state-of-the-art classical baselines. As illustrative cases, on four 50-band splits selected from Indian Pines, the quantum model achieved a 78.0±6.2% accuracy for a binary classification task compared to 72.0±5.0% for the standard radial basis function kernel. For a four-class classification task, the quantum kernel reached 83.3±3.1% accuracy, outperforming several state-of-the-art baselines. On five 75-band splits selected from the methane detection dataset, the quantum approach yielded 58.5±5.0% accuracy versus 55.1±2.5% for the classical counterpart. This study provides a foundation for future exploration of quantum machine learning in high-dimensional Earth observation tasks.