Quantum kernel methods have been actively examined from both theoretical and practical perspectives due to the potential of quantum advantage in machine learning tasks. Despite a provable advantage of fine-tuned quantum kernels for specific problems, widespread practical usage of quantum kernel methods requires resolving the so-called vanishing similarity issue, where exponentially vanishing variance of the quantum kernels causes implementation infeasibility and trainability problems. In this work, we analytically and numerically investigate the vanishing similarity issue in projected quantum kernels with alternating layered Ans\"atze. We find that variance depends on circuit depth, the size of local unitary blocks, and the initial state, indicating the issue is avoidable if shallow alternating layered Ans\"atze are used and the initial state is not highly entangled. Our work provides some insight into design principles of projected quantum kernels and implies the need for caution when using highly entangled states as input to quantum kernel-based learning models.
Quantum kernel methods have been actively examined from both theoretical and practical perspectives due to the potential of quantum advantage in machine learning tasks. Despite a provable advantage of fine-tuned quantum kernels for specific problems, widespread practical usage of quantum kernel methods requires resolving the so-called vanishing similarity issue, where exponentially vanishing variance of the quantum kernels causes implementation infeasibility and trainability problems. In this work, we analytically and numerically investigate the vanishing similarity issue in projected quantum kernels with alternating layered Ans"atze. We find that variance depends on circuit depth, the size of local unitary blocks, and the initial state, indicating the issue is avoidable if shallow alternating layered Ans"atze are used and the initial state is not highly entangled. Our work provides some insight into design principles of projected quantum kernels and implies the need for caution when using highly entangled states as input to quantum kernel-based learning models.
Quantum kernel methods have been actively examined from both theoretical and practical perspectives due to the potential of quantum advantage in machine learning tasks. Despite a provable advantage of fine-tuned quantum kernels for specific problems, widespread practical usage of quantum kernel methods requires resolving the so-called vanishing similarity issue, where exponentially vanishing variance of the quantum kernels causes implementation infeasibility and trainability problems. In this work, we analytically and numerically investigate the vanishing similarity issue in projected quantum kernels with alternating layered ansatzes. We find that variance depends on circuit depth, size of local unitary blocks and initial state, indicating the issue is avoidable if shallow alternating layered ansatzes are used and initial state is not highly entangled. Our work provides some insights into design principles of projected quantum kernels and implies the need for caution when using highly entangled states as input to quantum kernel-based learning models.
Abstract Quantum kernel (QK) methods exploit quantum computers to calculate QKs for the use of kernel-based learning models. Despite a potential quantum advantage of the method, the commonly used fidelity-based QK suffers from a detrimental issue, which we call the vanishing similarity issue; the exponential decay of the expectation value and the variance of the QK deteriorates implementation feasibility and trainability of the model with the increase of the number of qubits. This implies the need to design QKs alternative to the fidelity-based one. In this work, we propose a new class of QKs called the quantum Fisher kernels (QFKs) that take into account the geometric structure of the data source. We analytically and numerically demonstrate that the QFK can avoid the issue when shallow alternating layered ansatzes are used. In addition, the Fourier analysis numerically elucidates that the QFK can have the expressivity comparable to the fidelity-based QK. Moreover, we demonstrate synthetic classification tasks where QFK outperforms the fidelity-based QK in performance due to the absence of vanishing similarity. These results indicate that QFK paves the way for practical applications of quantum machine learning toward possible quantum advantages.
The hybrid quantum-classical algorithm is actively examined as a technique applicable even to intermediate-scale quantum computers. To execute this algorithm, the hardware efficient ansatz is often used, thanks to its implementability and expressibility; however, this ansatz has a critical issue in its trainability in the sense that it generically suffers from the so-called gradient vanishing problem. This issue can be resolved by limiting the circuit to the class of shallow alternating layered ansatz. However, even though the high trainability of this ansatz is proved, it is still unclear whether it has rich expressibility in state generation. In this paper, with a proper definition of the expressibility found in the literature, we show that the shallow alternating layered ansatz has almost the same level of expressibility as that of hardware efficient ansatz. Hence the expressibility and the trainability can coexist, giving a new designing method for quantum circuits in the intermediate-scale quantum computing era.
Abstract Quantum kernels (QKs) are widely used in quantum machine learning applications; yet, their potential to surpass classical machine learning methods on classical datasets remains uncertain. This limitation can be attributed to the exponential concentration phenomenon, which can impair generalization. A common strategy to alleviate this is bandwidth tuning, which involves rescaling data points in the quantum model to improve generalization. In this work, we numerically demonstrate that optimal bandwidth tuning results in QKs that closely resemble radial basis function (RBF) kernels, leading to a lack of quantum advantage over classical methods. Moreover, we reveal that the size of optimal bandwidth tuning parameters further simplifies QKs, causing them to behave like polynomial kernels, corresponding to a low-order Taylor approximation of a RBF kernel. We thoroughly investigate this for fidelity quantum kernels and projected quantum kernels using various data encoding circuits across several classification datasets. We provide numerical evidence and derive a simple analytical model that elucidates how bandwidth tuning influences key quantities in classification tasks. Overall, our findings shed light on the mechanisms that render QK methods classically tractable.
Quantum kernel methods promise enhanced expressivity for learning structured data, but their usefulness has been limited by kernel concentration and barren plateaus. Both effects are mathematically equivalent and suppress trainability. We analytically prove that covariant quantum kernels tailored to datasets with group symmetries avoid exponential concentration, ensuring stable variance and guaranteed trainability independent of system size. Our results extend beyond prior two-coset constructions to arbitrary coset families, broadening the scope of problems where quantum kernels can achieve advantage. We further derive explicit bounds under coherent noise models - including unitary errors in fiducial state preparation, imperfect unitary representations, and perturbations in group element selection - and show through numerical simulations that the kernel variance remains finite and robust, even under substantial noise. These findings establish a family of quantum learning models that are simultaneously trainable, resilient to coherent noise, and linked to classically hard problems, positioning group-symmetric quantum kernels as a promising foundation for near-term and scalable quantum machine learning.
The quantum kernel method is one of the key approaches to quantum machine learning, which has the advantage of not requiring optimization and its theoretical simplicity. By virtue of these properties, several experimental demonstrations and discussions of the potential advantages have been developed so far. However, as is the case in classical machine learning, not all quantum machine learning models could be regarded as kernel methods. In this work, we explore a quantum machine learning model with a deep parametrized quantum circuit and aim to go beyond the conventional quantum kernel method. In this case, the expressive power and performance are expected to be enhanced, while the training process might be a bottleneck because of the barren plateaus. Moreover, the high computational cost of gradient-based optimization and the large search space of the gradient-free optimization directly make the training intractable. However, we find that parameters of a deep enough quantum circuit do not move much from their initial values during training, allowing for a first-order expansion with respect to the parameters. This behavior is similar to that of the neural tangent kernel in classical machine learning, and such a quantum machine learning with deep variational quantum circuits can be described by another emergent kernel, the . We show that the proposed quantum tangent kernel has the potential to outperform the conventional quantum kernel method by performing a classification task on an ansatz-generated dataset. This work provides a different direction beyond the conventional quantum kernel method and explores the potential power of quantum machine learning with deep parametrized quantum circuits. Published by the American Physical Society 2024