Quantum kernel methods (QKMs) offer a potential speedup in machine learning by leveraging quantum computation to map data into high-dimensional Hilbert spaces, creating more expressive kernel functions than classical methods. While successful in small-scale experiments on synthetic data, scaling to large circuits on noisy hardware remains a challenge. This research addresses this limitation by experimentally implementing a QKM classifier on real-world high-dimensional data from cosmology. Previous experiments often used artificial or heavily pre-processed data, limited qubit numbers, or circuit connectivity unsuitable for Noisy Intermediate-Scale Quantum (NISQ) processors. This work overcomes these challenges by using up to 17 qubits on the Google Sycamore processor, employing a circuit design that maximizes kernel magnitudes and mitigating errors specific to QKM calculations on NISQ devices. The dataset used is the Photometric LSST Astronomical Time-series Classification Challenge (PLASTICC) data, focusing on a binary classification problem distinguishing between Type Ia and Type II supernovae. The 67-dimensional feature vectors extracted from the time-series data are directly used without dimensionality reduction. A key goal is to assess the performance of QKMs on real, high-dimensional data without extensive classical preprocessing, pushing the boundaries of NISQ applications in machine learning. Supervised learning, specifically Support Vector Machines (SVMs), provides the classical framework. The kernel trick allows SVMs to handle non-linearly separable data by using kernel functions representing inner products in high-dimensional feature spaces. QKMs offer the potential to improve upon this by using a quantum circuit to map data into a complex Hilbert space, enabling more complex and potentially classically intractable kernels. The specific QKM approach in this work involves using a quantum circuit U(x) to map real data x into quantum state |ψ(x)> = U(x)|0>, where the kernel is defined as the squared inner product |<ψ(x)|ψ(x)>|^2, enabling more expressive models than simply using <ψ(x)|ψ(x)>. The experimental setup involves sampling the quantum circuit outputs to estimate the kernel matrix K, which is then used to train a classical SVM. The experiment’s success hinges on designing a circuit that produces large kernel magnitudes to minimize statistical errors and appropriately maps data to separate classes.

The paper reviews prior work in quantum kernel methods, highlighting limitations in existing implementations that primarily focused on small-scale experiments with synthetic or heavily pre-processed data, often utilizing a limited number of qubits or circuit designs unsuitable for NISQ processors. The authors note recent progress in applying QKMs to high-energy physics data but emphasize the need for further exploration of high-dimensional data on noisy hardware. The literature review emphasizes the potential of QKMs to offer a speedup over classical approaches for specific data classes but also acknowledge the lack of thorough investigation into scaling to larger circuits on noisy hardware and the need for more robust error mitigation techniques.

The study utilizes data from the PLASTICC challenge, focusing on a binary classification task distinguishing between Type Ia and Type II supernovae. The data is pre-processed to create 67-dimensional feature vectors for each object, which are then mapped into a quantum state using a designed quantum circuit. This circuit, depicted in Figure 2, employs a series of single-qubit rotations parameterized by the data features followed by entangling gates (iSWAP). The choice of circuit is motivated by its ability to produce large kernel matrix elements which minimizes statistical errors during sampling on the noisy quantum computer. Experiments were conducted on Google's Sycamore processor, using several qubit counts (10, 14, and 17 qubits). To reduce the impact of hardware noise, the study employs various error mitigation techniques including a readout error correction method. The kernel matrix is empirically estimated by sampling the output of the quantum circuit for each pair of data points a sufficient number of times (5000 repetitions). This kernel matrix is then used to train a classical SVM classifier using a training set. The performance is evaluated on a separate test set to avoid overfitting. The optimal SVM hyperparameter (C) is determined through leave-one-out cross-validation (LOOCV) on the training set, ensuring robustness against noise. The learning curve (Figure 3) for varying training set sizes is established through noiseless simulation to help choose an appropriate training and test data size for the hardware experiments, aiming to balance model complexity and generalization performance. The final hardware experiment utilizes a balanced training and test set selected to minimize variance in classification scores for small test sets, ensuring that the reported hardware performance is not inflated due to chance. The parallel execution of entangling gates is implemented to improve efficiency, and the chosen number of qubits for each experiment is selected based on device connectivity and calibration data.

The experimental results demonstrate that the quantum kernel classifier achieves a test set accuracy comparable to noiseless simulations across different numbers of qubits (10, 14, and 17), even with significant hardware noise that suppresses observed bitstring probabilities. Figure 4 illustrates the classifier's robustness to noise, achieving competitive performance despite the reduction in kernel magnitudes due to the imperfect fidelity of the quantum gates. The experiment indicates that the performance of the QKM is not fundamentally limited by the number of qubits used within this range, showing reasonably high accuracy even with up to 70% suppression of kernel magnitudes due to noise. This robustness is attributed to the SVM's invariance under scaling transformations of the kernel matrix. The experiment successfully leverages tens of qubits for a real-world classification task on high-dimensional data, highlighting the potential of NISQ processors. The large fraction of support vectors selected in both simulated and hardware experiments (around 90%) is noted. This suggests a complex decision boundary and the significant influence of noise in the hardware-generated kernel matrix.

The study successfully demonstrates the feasibility of performing machine learning on high-dimensional real-world data using a NISQ processor. The achieved accuracy, comparable to noiseless simulations despite hardware noise and the absence of quantum error correction, is significant. The results highlight that QKMs can operate effectively on near-term devices, despite not currently achieving quantum advantage. The research emphasizes the importance of careful consideration of shot statistics and kernel magnitude when evaluating QKM performance. The robustness of the classifier to noise and the scalability across different qubit counts are crucial observations. The study motivates future research to find datasets where quantum machine learning might outperform classical approaches, suggesting the investigation of datasets reflecting intrinsically complex correlations difficult to represent classically, possibly involving quantum data from quantum many-body systems or quantum sensing applications.

This work demonstrates the successful application of a quantum kernel classifier to a real, high-dimensional dataset using a NISQ device. The achieved accuracy, comparable to noiseless simulations despite significant hardware noise, suggests QKMs can be practical on near-term quantum computers. Future research should focus on identifying datasets where quantum machine learning demonstrates a clear advantage over classical methods, possibly utilizing intrinsically complex quantum data sources.

The study acknowledges that the implemented circuits do not currently demonstrate quantum advantage over classical approaches. While error mitigation techniques were used, the impact of hardware noise is still a limiting factor. The reliance on classical SVM training and post-processing restricts the full potential of quantum computation. The relatively large number of support vectors observed suggests a potential for improvement in algorithm efficiency. Further investigation into the impact of different error sources and the development of more efficient error mitigation strategies are necessary.