Towards understanding the power of quantum kernels in the NISQ era

Abstract

A key problem in the field of quantum computing is understanding whether quantum machine learning (QML) models implemented on noisy intermediate-scale quantum (NISQ) machines can achieve quantum advantages. Recently, Huang et al. [Nat Commun 12, 2631] partially answered this question by the lens of quantum kernel learning. Namely, they exhibited that quantum kernels can learn specific datasets with lower generalization error over the optimal classical kernel methods. However, most of their results are established on the ideal setting and ignore the caveats of near-term quantum machines. To this end, a crucial open question is: does the power of quantum kernels still hold under the NISQ setting? In this study, we fill this knowledge gap by exploiting the power of quantum kernels when the quantum system noise and sample error are considered. Concretely, we first prove that the advantage of quantum kernels is vanished for large size of datasets, few number of measurements, and large system noise. With the aim of preserving the superiority of quantum kernels in the NISQ era, we further devise an effective method via indefinite kernel learning. Numerical simulations accord with our theoretical results. Our work provides theoretical guidance of exploring advanced quantum kernels to attain quantum advantages on NISQ devices.

Biology and medicine in the landscape of quantum advantages

“…Many of the quantum advantages associated with near term variational QML algorithms relate to model capacity, expressivity, and sample efficiency. In particular, variational QML algorithms may yield reductions in the number of required trainable parameters [238], generalization error [226,227,228,229], the number of examples required to learn a model [236,222], and improvements in training landscapes [277,226,227,231,232]. Evidence supporting one or more of these advantages has been found in both theoretical models and proof of principle implementations of quantum neural networks (QNNs) [226,227,228,231] and quantum kernel methods (QKM) [222,229,225].…”

“…In particular, variational QML algorithms may yield reductions in the number of required trainable parameters [238], generalization error [226,227,228,229], the number of examples required to learn a model [236,222], and improvements in training landscapes [277,226,227,231,232]. Evidence supporting one or more of these advantages has been found in both theoretical models and proof of principle implementations of quantum neural networks (QNNs) [226,227,228,231] and quantum kernel methods (QKM) [222,229,225]. It is notable that these methods are both closely related to VQAs leveraging gradient-based classical optimizers (indeed, they often share overlapping definitions in the literature, as briefly noted in 2019 [213]) [223,411,412].…”

“…Advantages in generalization error are dependent on a variety of factors, including the encoding used (with basis encoding performing particularly poorly [416]) and the availability of data sufficient to train a comparable classical model [225]. While there is substantial evidence supporting reductions in generalization error [226,227,228,229], evidence of poor generalization performance under certain constructions also exists (e.g. see this recent paper [417]).…”

“…While many variational QML methods lack analytical bounds on their time complexity (unlike many of the earlier QML algorithms), they may nonetheless offer other forms of quantum advantage. In particular, theoretical and empirical evidence exist for improvements in generalization error [226,227,228,229], trainability (i.e. with certain constructions, favorable training landscapes with fewer barren plateaus and narrow gorges [226,227,230,231,232,233]), and sample complexity [234,235,236].…”

Unentangled quantum reinforcement learning agents in the OpenAI Gym

“…Concretely, for the synthetic dataset, Refs. [53][54][55] exhibited the advantages of quantum neural networks [56,57] and quantum kernels [56] in the measure of generalization error [58][59][60][61][62][63]. However, quantum supervised and unsupervised learning models may encounter trainability issues, where the gradients exponentially vanish for the number of qubits [64,65].…”