Kernel methods are a cornerstone of classical machine learning. The idea of using quantum computers to compute kernels has recently attracted attention. Quantum embedding kernels (QEKs), constructed by embedding data into the Hilbert space of a quantum computer, are a particular quantum kernel technique that is particularly suitable for noisy intermediate-scale quantum devices. Unfortunately, kernel methods face three major problems: Constructing the kernel matrix has quadratic computational complexity in the number of training samples, choosing the right kernel function is nontrivial, and the effects of noise are unknown. In this work, we addressed the latter two. In particular, we introduced the notion of trainable QEKs, based on the idea of classical model optimization methods. To train the parameters of the QEK, we proposed the use of kernel-target alignment. We verified the feasibility of this method, and showed that for our experimental setup we could reduce the training error significantly. Furthermore, we investigated the effects of device and finite sampling noise, and we evaluated various mitigation techniques numerically on classical hardware. We took the best performing strategy and evaluated it on data from a real quantum processing unit. We found that using this mitigation strategy demonstrated an increased kernel matrix quality.
Kernel methods are a cornerstone of classical machine learning. The idea of using quantum computers to compute kernels has recently attracted attention. Quantum embedding kernels (QEKs), constructed by embedding data into the Hilbert space of a quantum computer, are a particular quantum kernel technique that is particularly suitable for noisy intermediate-scale quantum devices. Unfortunately, kernel methods face three major problems: Constructing the kernel matrix has quadratic computational complexity in the number of training samples, choosing the right kernel function is nontrivial, and the effects of noise are unknown. In this work, we addressed the latter two. In particular, we introduced the notion of trainable QEKs, based on the idea of classical model optimization methods. To train the parameters of the QEK, we proposed the use of kernel-target alignment. We verified the feasibility of this method, and showed that for our experimental setup we could reduce the training error significantly. Furthermore, we investigated the effects of device and finite sampling noise, and we evaluated various mitigation techniques numerically on classical hardware. We took the best performing strategy and evaluated it on data from a real quantum processing unit. We found that using this mitigation strategy demonstrated an increased kernel matrix quality.
In recent years, quantum computers have emerged as promising candidates for implementing kernels. Quantum Embedding Kernels embed data points into quantum states and calculate their inner product in a high-dimensional Hilbert Space by computing the overlap between the resulting quantum states. Variational Quantum Circuits (VQCs) are typically used for this end, with Kernel Target Alignment (KTA) as cost function. The optimized kernels can then be deployed in Support Vector Machines (SVMs) for classification tasks. However, both classical and quantum SVMs scale poorly with increasing dataset sizes. This issue is exacerbated in quantum kernel methods, as each inner product requires a quantum circuit execution. In this paper, we investigate KTA-trained quantum embedding kernels and employ a low-rank matrix approximation, the Nyström method, to reduce the quantum circuit executions needed to construct the Kernel Matrix. We empirically evaluate the performance of our approach across various datasets, focusing on the accuracy of the resulting SVM and the reduction in quantum circuit executions. Additionally, we examine and compare the robustness of our model under different noise types, particularly coherent and depolarizing noise.
Quantum kernel methods offer significant theoretical benefits by rendering classically inseparable features separable in quantum space. Yet, the practical application of Quantum Machine Learning (QML), currently constrained by the limitations of Noisy Intermediate-Scale Quantum (NISQ) hardware, necessitates effective strategies to compress and embed large-scale real-world data like images into the constrained capacities of existing quantum devices or simulators. To this end, we propose Quantum Generator Kernels (QGKs), a generator-based approach to quantum kernels, comprising a set of Variational Generator Groups (VGGs) that merge universal generators into a parameterizable operator, ensuring scalable coverage of the available quantum space. Thereby, we address shortcomings of current leading strategies employing hybrid architectures, which might prevent exploiting quantum computing's full potential due to fixed intermediate embedding processes. To optimize the kernel alignment to the target domain, we train a weight vector to parameterize the projection of the VGGs in the current data context. Our empirical results demonstrate superior projection and classification capabilities of the QGK compared to state-of-the-art quantum and classical kernel approaches and show its potential to serve as a versatile framework for various QML applications.
Quantum computing (QC) seems to show potential for application in machine learning (ML). In particular quantum kernel methods (QKM) exhibit promising properties for use in supervised ML tasks. However, a major disadvantage of kernel methods is their unfavorable quadratic scaling with the number of training samples. Together with the limits imposed by currently available quantum hardware (NISQ devices) with their low qubit coherence times, small number of qubits, and high error rates, the use of QC in ML at an industrially relevant scale is currently impossible. As a small step in improving the potential applications of QKMs, we introduce QUACK, a quantum kernel algorithm whose time complexity scales linear with the number of samples during training, and independent of the number of training samples in the inference stage. In the training process, only the kernel entries for the samples and the centers of the classes are calculated, i.e. the maximum shape of the kernel for n samples and c classes is (n, c). During training, the parameters of the quantum kernel and the positions of the centroids are optimized iteratively. In the inference stage, for every new sample the circuit is only evaluated for every centroid, i.e. c times. We show that the QUACK algorithm nevertheless provides satisfactory results and can perform at a similar level as classical kernel methods with quadratic scaling during training. In addition, our (simulated) algorithm is able to handle high-dimensional datasets such as MNIST with 784 features without any dimensionality reduction.
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 machine learning with quantum kernels for classification problems is a growing area of research. Recently, quantum kernel alignment techniques that parameterise the kernel have been developed, allowing the kernel to be trained and therefore aligned with a specific dataset. While quantum kernel alignment is a promising technique, it has been hampered by considerable training costs because the full kernel matrix must be constructed at every training iteration. Addressing this challenge, we introduce a novel method that seeks to balance efficiency and performance. We present a sub-sampling training approach that uses a subset of the kernel matrix at each training step, thereby reducing the overall computational cost of the training. In this work, we apply the sub-sampling method to synthetic datasets and a real-world breast cancer dataset and demonstrate considerable reductions in the number of circuits required to train the quantum kernel while maintaining classification accuracy.
Quantum and classical machine learning have been naturally connected through kernel methods, which have also served as proof-of-concept for quantum advantage. Quantum embeddings encode classical data into quantum feature states, enabling the construction of embedding quantum kernels (EQKs) by measuring vector similarities and projected quantum kernels (PQKs) through projections of these states. However, in both approaches, the model is influenced by the choice of the embedding. In this work, we propose using the training of a quantum neural network (QNN) to construct neural quantum kernels, specifically neural EQKs and neural PQKs—problem-inspired kernel functions. Unlike previous approaches, our method requires the kernel matrix to be constructed only once, significantly reducing computational overhead. To achieve this, we introduce a scalable training method for an n -qubit data reuploading QNN. Furthermore, we demonstrate neural quantum kernels can alleviate exponential concentration and enhance generalization capabilities compared to problem-agnostic kernels, positioning them as a scalable and robust solution for quantum machine learning applications.