Chapter 3.4 Recent Advancements

The foundational concept of using quantum computers to evaluate kernel functions, namely the concept of quantum kernels, was first explored by @schuld2017implementing, who highlighted the fundamental differences between quantum kernels and quantum support vector machines. Building on this, @havlivcek2019supervised and @schuld2019quantum established a connection between quantum kernels and parameterized quantum circuits (PQCs), demonstrating their practical implementation. These works emphasized the parallels between quantum feature maps and the classical kernel trick. Since then, a large number of studies delved into figuring out the potential of quantum kernels for solving practical real-world problems.

The recent advancements in quantum kernel machines can be roughly categorized into three key areas: kernel design, theoretical findings, and applications. Specifically, the advances in kernel design focus on addressing challenges such as vanishing similarity and kernel concentration by exploring innovative frameworks. Theoretical studies delve into the limitations and capabilities of quantum kernels, examining factors such as generalization error bounds, noise resilience, and their capacity to demonstrate quantum advantage. Finally, applications of quantum kernels showcase their potential across diverse domains. In the rest of this section, we separately review the existing developments within each of these three branches.

Quantum kernel design

A crucial research line in this field focuses on constructing trainable quantum kernels to maximize performance for specific datasets and problem domains. In particular, traditional quantum kernels, with fixed data embedding schemes, are limited to specific feature representation spaces and often fail to capture the complex and diverse patterns inherent in real-world data. To address this limitation, @lloyd2020quantum explored the construction of task-specific quantum feature maps using measurement theory. Building on this idea, @hubregtsen2022training introduced a method to optimize quantum feature maps variationally within quantum kernels, linking this approach to the concept of data re-uploading techniques [@perez2020data; @schuld2021effect]. Additionally, @vedaie2020quantum and @lei2024neural proposed leveraging multiple kernel learning and architecture search to construct quantum kernels, respectively. Last, @glick2024covariant proposed the covariant quantum kernels for efficiently solving the problems with group structure.

An orthogonal research direction is addressing the issue of exponential kernel concentration, also known as vanishing similarity, in quantum kernels [@thanasilp2022exponential]. Specifically, quantum kernels, which are defined as the inner product between quantum feature mappings, often suffer from the phenomenon of vanishing similarity. This issue was first highlighted by @huang2021power, who found that quantum feature mappings are typically “far” from one another in high-dimensional feature spaces, leading to vanishing similarity and, consequently, poor generalization performance.

To mitigate this issue, @huang2021power introduced projected quantum kernels, which store feature vectors in classical memory and evaluate a Gaussian kernel. This approach replaces the reliance on the inner product of quantum states, as seen in traditional quantum embedding kernels. Moreover, @suzuki2022quantum proposed the anti-symmetric logarithmic derivative quantum Fisher kernel, which avoids the exponential kernel concentration problem by encoding geometric information of the input data. Beyond developing new types of quantum kernels, @shaydulin2022importance and @canatar2022bandwidth explored strategies to mitigate the exponential concentration issue for quantum embedding kernels by scaling input data with carefully chosen hyperparameters. This approach clusters the data-encoded quantum states closer together in feature space, reducing the risk of vanishing similarity at the cost of slightly lowering expressivity.

Theoretical studies of quantum kernels

The theoretical studies of quantum kernels aim to rigorously understand their performance potential and limitations under realistic conditions, enabling the design of more effective, robust, and generalizable quantum kernel methods. Prior literature in this context focuses on exploring two aspects of quantum kernels, namely, expressivity and generalization ability.

Expressivity of quantum kernels

The expressivity of quantum kernels refers to their capacity to capture complex data relationships and represent intricate patterns in the feature space. A common approach to studying this is through the analysis of the reproducing kernel Hilbert space (RKHS), which provides insights into the underlying feature representations of quantum kernels.

@schuld2021supervised rigorously analyzed the RKHS of embedding-based quantum kernels and established the universality approximation theorem, demonstrating that quantum kernels can approximate a wide class of functions. Building on this, @jerbi2023quantum extended the analysis by investigating parameterized quantum embedding kernels, introducing a data-reuploading structure and proving a corresponding universality approximation theorem. These results underscore the expressive power of quantum kernels in representing complex data structures.

Despite these advances, the studies on expressive power and universality approximation often overlook the efficiency of constructing quantum kernels. Specifically, if the computational cost of constructing a universal quantum kernel is comparable to that of classical methods, the practical advantages of quantum kernels become questionable.

To narrow this gap, @gil2024expressivity examined the expressive power of efficient quantum kernels that can be implemented on quantum computers within polynomial time. Their work provides a detailed analysis of the types of kernels that are achievable with a polynomial number of qubits and within polynomial time, offering insights into the feasibility and practical utility of quantum kernels in real-world scenarios.

However, alongside the exploration of expressive power, a significant challenge-known as exponential kernel concentration-has been identified. @thanasilp2022exponential identified four key factors contributing to this issue: high expressivity of data embeddings, global measurements, entanglement, and noise. To address this limitation, substantial research has focused on designing advanced quantum kernels to mitigate exponential kernel concentration, as discussed in Chapter 1.5.1{reference-type=“ref” reference=“chapt3:subsec:qk_design”}.

Generalization of quantum kernels

The generalization ability of a learning model—its capacity to perform well on unseen data—is a critical factor in evaluating its effectiveness. In this context, a considerable body of research has investigated the generalization ability of quantum kernels. @huang2021power established a data-dependent generalization error bound for quantum kernels and demonstrated that, for certain types of data (such as those generated by quantum circuits), quantum kernels can achieve a generalization advantage over classical learning models.

In addition, @wang2021towards explored the generalization performance of quantum kernels in the noisy scenario, where practical limitations such as finite measurements and quantum noise are taken into account. Their work rigorously showed that the generalization performance of quantum kernels could be significantly degraded in scenarios involving large training datasets, limited measurement repetitions, or high levels of system noise. To address these challenges, they proposed an effective method based on indefinite kernel learning to help preserve generalization performance under such constraints.

Beyond quantum data, @liu2021rigorous examined the generalization error of quantum kernels using artificial classical datasets, such as those based on the discrete logarithm problem. Their results demonstrated that quantum kernels could achieve accurate predictions in polynomial time for such problems, whereas classical learning models require exponential time, highlighting the potential computational advantages of quantum kernels.

Despite these promising results, @kubler2021inductive studied the generalization ability of quantum kernels from the perspective of inductive bias. They argued that quantum kernels, lacking inductive bias, often fail to outperform classical models in practical scenarios. This underscores the importance of carefully designing embeddings and aligning kernels to achieve meaningful and practical quantum advantages.

Provable advantages of quantum kernels

The potential for quantum kernels to demonstrate quantum advantage has been a central focus of research. For instance, @huang2021power provided evidence of generalization advantages for quantum kernels on quantum data. Similarly, @liu2021rigorous presented a rigorous framework showing that quantum kernels can efficiently solve problems like the discrete logarithm problem, which is believed to be intractable for classical computers under standard cryptographic assumptions. Moreover, @sweke2021quantum demonstrated quantum advantage in distribution learning tasks, offering some of the earliest theoretical evidence of quantum advantage in machine learning.

However, many of these tasks are artificial, designed specifically to showcase quantum advantages. This raises the question of how these theoretical benefits can be translated to real-world applications. In this regard, the next significant challenge is to demonstrate that quantum models can consistently outperform classical models in solving practical, real-world problems.

Applications of quantum kernels

Motivated by the potential of quantum kernels to recognize complex data patterns, numerous studies have explored their practical applications across diverse fields, including classification, drug discovery, anomaly detection, and financial modeling.

For instance, @beaulieu2022quantum investigate the use of quantum kernels for image classification, specifically in identifying real-world manufacturing defects. Similarly, @rodriguez2024satellite apply quantum kernels to satellite image classification, a task of particular importance in the earth observation industry. In the field of quantum physics, @sancho2022quantum and @wu2023quantum leverage quantum kernels to recognize phases of quantum matter, where quantum kernels outperform classical learning models in solving certain problems. In drug discovery, @batra2021quantum explore the potential of quantum kernels to accelerate and improve the identification of promising compounds.

Quantum kernels have also been explored in anomaly detection. @liu2018quantum demonstrate their superior performance over classical methods in detecting anomalies within quantum data. Furthermore, @grossi2022mixed employ quantum kernel methods for fraud classification tasks, showing improvements when benchmarked against classical methods. @miyabe2023quantum expand their application to the financial domain by proposing a quantum multiple-kernel learning methodology. This approach broadens the scope of quantum kernels to include credit scoring and directional forecasting of asset price movements, highlighting their potential utility in financial services.

Despite the promise shown in these applications, the realization of quantum advantage in practical tasks remains an ongoing area of research, with current efforts directed toward identifying real-world problems where quantum kernels outperform classical alternatives.