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arXiv:2210.16581 · Quantum Science and Technology

Quantum Fisher kernel for mitigating the vanishing similarity issue

Yudai Suzuki, Hideaki Kawaguchi, Naoki Yamamoto

Published May 14, 2024• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2058-9565/ad4b97

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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.
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 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 support vector machines are classification algorithms that rely on quantum-generated kernels. The fidelity quantum kernel commonly used in quantum support vector machines suffers from exponential concentration as system size increases, preventing an efficient scaling beyond fewqubit systems. We introduce the Hamming quantum kernel, a classical post-processing method that is based on the same measurement outcomes as the fidelity quantum kernel. However, it avoids the exponential concentration problem by using the full measurement statistics rather than a single fidelity value. We evaluate the approach on both classical data (MNIST) and synthetic data generated from quantum circuits, using systems ranging from 2 to 27 qubits. Throughout the simulations, the Hamming quantum kernel outperforms the fidelity quantum kernel whenever 15 or more qubits are used. Furthermore, for synthetic quantum data, our method consistently outperforms the classical Gaussian kernel. This demonstrates that the Hamming quantum kernel improves the expressivity and robustness at larger qubit scales without requiring any additional quantum ressources.
Quantum machine learning techniques are commonly considered one of the most promising candidates for demonstrating practical quantum advantage. In particular, quantum kernel methods have been demonstrated to be able to learn certain classically intractable functions efficiently if the kernel is well aligned with the target function. In the more general case, quantum kernels are known to suffer from exponential ``flattening'' of the spectrum as the number of qubits grows, preventing generalization and necessitating the control of the inductive bias by hyperparameters. We show that the general-purpose hyperparameter-tuning techniques proposed to improve the generalization of quantum kernels lead to the kernel becoming well approximated by a classical kernel, removing the possibility of quantum advantage. We provide extensive numerical evidence for this phenomenon utilizing multiple previously studied quantum feature maps and both synthetic and real data. Our results show that unless novel techniques are developed to control the inductive bias of quantum kernels, they are unlikely to provide a quantum advantage on classical data that lacks special structure.
This artifact includes the software we have developed to conduct an experimental analysis of QFK, providing its first implementation with real-world datasets and exploring its behavior with different quantum circuit architectures. Wedemonstrate that QFK is capable of creating models useful for learning and that it can be effectively executed on currentNISQ devices, successfully mitigating exponential concentration and offering high accuracy both in ideal and noisy simulations. Furthermore, we show that QFK can compete with state-of-the-art kernel methods, offering promising scalability even in settings involving an increasing number of qubits and highly expressive quantum circuits. This is an initial demonstration that QFK may offer a quantum advantage in learning tasks, making it a competitive alternative to the standard approaches.