Quantum-Enhanced Support Vector Machine for Large-Scale Stellar Classification with GPU Acceleration

The elucidation of stellar classifications hinges upon the analysis of stellar spectra, which provides insights into the temperature, luminosity, and chemical makeup of stars [13]. At the core of this scientific endeavour lies the Harvard spectral classification system, an enduring schema that categorizes stars into seven primary types (O, B, A, F, G, K, M), shown in Fig. 1. These classifications draw from the seminal work of Annie Jump Cannon [14], whose early 20th-century schematics continue to underpin modern astrophysics.

Spectral types serve as proxies for a star’s thermal and luminous signatures, with O-type stars being the most incandescent and luminous, descending through to the cooler and dimmer M-type stars. Beyond mere temperature categorization, the luminosity class of a star offers a window into its dimensional attributes and stage of evolution, ranging from the expansive supergiants (Class I) to the more diminutive main sequence stars (Class V) [15]. The Hertzsprung-Russell (HR) diagram, an indispensable chart plotting stellar luminosity against surface temperature, further refines our understanding of these celestial bodies [16]. The HR diagram demarcates distinct domains occupied by stars at various junctures of mass, age, and evolution, offering a macroscopic view of stellar populations. Figure 1 in the present document exemplifies such HR diagram, assigning colours to stars based on their spectral classification, with the bluest hues representing the torrid O stars and the reddest shades corresponding to the temperate M stars.

Traditionally, the classification of stars has been a manual process, contingent on the meticulous examination of spectral lines, a practice that is not only laborious but also prone to inconsistency [17]. The advent of machine learning, particularly the deployment of SVMs, has revolutionized this task by enabling a more automated and objective analysis, thereby mitigating human error [18]. Despite these advances, the burgeoning complexity and sheer volume of astronomical datasets are challenging the capabilities of conventional SVMs. It is within this context that quantum-enhanced SVMs emerge as a novel paradigm, promising to surpass the limitations of existing techniques and revolutionize the precision and computational efficiency of stellar classification [19].

In this exploration, we delve into the potential of such quantum-augmented analytical methodologies to not only refine but also accelerate the classification processes that are pivotal to our comprehension of the stellar population.

Machine learning is a powerful tool for solving complex problems in fields ranging from computer vision to natural language processing. One of the most popular algorithms within this domain is the SVM, which is particularly effective for classification tasks. The advent of quantum computing has given rise to new possibilities in the field of machine learning, with QSVM being one of the most promising developments for classification problems.

In classical SVM, the algorithm processes a set of input data points along with their respective labels to find the optimal hyperplane that separates the data into two distinct classes [20]. The objective is to maximize the margin between these classes, which in turn informs the classification of new data points based on their position relative to the hyperplane, as shown in Fig. 2.

QSVM extends the principles of the classical SVM into the quantum realm. The process involves utilizing a quantum kernel function that maps input data points to a high-dimensional Hilbert space. This mapping allows for the computation of the inner product between the data points, from which the probability of class affiliation is derived for classification purposes.

One common quantum kernel is the Gaussian kernel, implemented in the quantum setting as follows [21]:

where $x_{i}$ and $x_{j}$ represent input data vectors and $\sigma$ is the kernel’s width parameter.

The principal advantage of QSVM over classical SVM is the utilization of quantum parallelism, an inherent attribute of quantum computing that enables the simultaneous computation of the kernel function across multiple data point pairs. This capability can theoretically lead to substantial computational speedups. For example, computing the kernel function for all $n^{2}$ pairs of data points could, in theory, be executed in $O(\log n)$ time on a quantum computer, which is significantly faster compared to the $O(n^{2})$ time required on a classical computer[22].

Additionally, QSVM is poised to exploit quantum entanglement to potentially surpass the classification performance of its classical counterpart. Entanglement, a unique quantum mechanical phenomenon, enables unprecedented correlations between particles that classical systems cannot replicate. By capitalizing on entanglement, QSVMs are expected to achieve greater classification accuracies, particularly beneficial in managing the large and intricate datasets commonly found in fields like astrophysics and high-energy physics[23].

The Star Categorization - Giants and Dwarfs dataset, which is accessible on the Kaggle platform, presents a large collection of stars and their stellar properties based on multiple catalogues[24, 25, 26]. This collection encompasses data from more than 100,000 stars, enumerating attributes such as luminosity, temperature, radius, and mass. The relation among each category, colour index and absolute magnitude can be examined by the HR diagram shown in Fig. 3.

Within this dataset, stars are systematically classified into two predominant groups: giants and dwarfs. Giants are characterized as stars that surpass the Sun in terms of both luminosity and size, whereas dwarfs are smaller and less luminous when compared with the Sun. The classification is deduced from the values of luminosity and radius, which are themselves derived from spectral data. This dataset is essential in the study of stellar classification, which utilizes spectral data to categorize stars into various categories. Predominantly based on the Morgan–Keenan (MK) classification system[27], it employs the historical HR classification system for chromaticity categorization and uses Roman numerals for sizing stars. Essential to this dataset is the absolute magnitude and B-V Color Index, which are pivotal in distinguishing giants and dwarfs, thus offering a comprehensive insight into the distinct characteristics of these celestial bodies.

In the context of this paper, the dataset serves as a foundational framework for examining various stellar attributes, as well as for the development of machine learning models that can effectively classify stars into giants or dwarfs according to their stellar parameters. Furthermore, it provides a benchmark for evaluating the efficacy of quantum machine learning algorithms within the domains of astronomy. This study using quantum machine learning would explore the possibility of utilizing the quantum algorithm in astronomy research in the generation of big data.

For the task of classifying stars as either white dwarfs or giants, QKL emerges as a particularly promising technique, given its potential for high precision in predictions. Nevertheless, the intrinsic characteristics of quantum data, coupled with the current limitations in quantum hardware access, elevate the importance of feature engineering within this domain. To optimize data for QKL, traditional preprocessing methods are employed to streamline the dataset’s dimensionality and to isolate pertinent features prior to engaging the quantum kernel. This strategy is aimed at diminishing the computational load and enhancing model accuracy by curtailing the noise and ambiguities inherent in quantum datasets.

One approach to readying the data for QKL involves the conversion of the quantum state into a representation more conducive to machine learning algorithms. This may entail the application of quantum circuits or alternative transformations to distil beneficial features. For instance, the B-V colour index, indicative of stellar temperature, could be encoded into a quantum state, given its significance in star classification. In a similar vein, the spectral type could be encoded to reflect the pattern of spectral lines within the star’s spectrum, a crucial classification determinant.

The feature selection process exerts a considerable influence on the quantum kernel’s accuracy, as the kernel’s role is to assess the similarity between data point pairs. The features chosen for comparison, therefore, are pivotal. Meticulous feature engineering is imperative to ensure the kernel is attuned to the data’s most informative attributes. For example, the visual apparent magnitude and absolute magnitude are likely to be critical for accurate star classification, while other attributes, such as $e\_Plx$, may lack relevance and could be excluded. The essence of feature engineering, therefore, is paramount to the efficacy of QKL in stellar classification tasks.

Understanding the intricate relationships among features is crucial for the classification. Graphical representations of these interactions provide insights into how different attributes, such as absolute magnitude and B-V colour index, influence the stellar classification (a dwarf or a giant), shown in Fig. 6 and Fig. 7.

Analysis of these figures reveals the efficacy of composite features in highlighting subtle variances across attributes, enhancing the capabilities for the classification. Particularly, such features facilitate a clearer separation between data clusters, thereby potentially increasing the accuracy of the classification.

The use of such composite parameters, as demonstrated in Fig. 8, could be crucial in addressing complex classification challenges by leveraging discernible patterns within the feature space. This strategy could effectively overcome obstacles encountered in the delineation between classes of stars.

QKL leverages the principles of quantum computing to enhance classical machine learning frameworks. At the core of QKL is a VQC that acts as a quantum feature map, mapping classical data into an expanded quantum feature space. The parameters of the VQC are fine-tuned using classical optimization algorithms to improve the feature space representation.

Within this framework, the ZZFeatureMap stands out as a specific implementation of a quantum feature map. It employs ZZ gates to intricately encode interactions between features onto quantum states. The quantum circuit is shown in Fig. 9. This method is particularly adept at capturing complex correlations within the data, thereby improving the performance of machine learning tasks, such as classification, by utilizing the enriched feature space inherent to quantum systems.

QKL has the potential to surpass classical machine learning algorithms by leveraging these quantum-enhanced feature spaces. Moreover, when QKL is integrated with advanced computational resources, such as Quantum Processing Units (QPUs) or GPUs, it can significantly boost computational efficiency and learning efficacy.

Our investigation into the application of QKL for star classification across large datasets is captured in Figure 10. The performance metrics, including accuracy, F1 score, specificity, and sensitivity, indicate an enhancement in the QKL’s predictive power with increasing training samples, especially beyond the 15,000-sample threshold.

As seen in the depicted results, QKL achieves superior accuracy, F1 score, and specificity when compared to its classical counterpart. Despite a marginal reduction in sensitivity, the quantum kernel upholds a consistent performance, maintaining a score near 0.9. This consistency underscores the robustness of QKL in classifying stellar objects. Moreover, the quantum kernel’s performance demonstrates greater stability relative to the classical kernel across a variety of scenarios. This stability and improved performance are likely due to the quantum kernel’s ability to process the complex features inherent in astronomical data more efficiently, resulting in enhanced prediction accuracy and stability.

Quantum computing holds the potential to solve complex problems in machine learning. However, training quantum models is computationally intensive and necessitates specialized hardware. QKL, a technique involving the pre-processing of data on classical hardware prior to training a quantum kernel, has demonstrated promising outcomes. CuQuantum, an NVIDIA software library, facilitates the training of quantum models on GPUs, thereby enhancing both the speed and accuracy of predictions. This acceleration stems from the GPUs’ parallel processing capabilities. Moreover, the integration of CuQuantum with CUDA reduces computational costs, rendering QKL more accessible and scalable. Benchmarking results, as depicted in Fig. 14, compare the performance of the QKL algorithm on an M1 Pro CPU and an A100 GPU utilizing the CuQuantum library. These results showcase a significant speedup, with the GPU implementation outperforming the CPU by a factor of three, particularly when the QKL algorithm handles four features. This comparison underscores the efficiency and performance benefits of leveraging GPUs for quantum machine learning tasks.