Quantum computing, with its potential to enhance various machine learning
tasks, allows significant advancements in kernel calculation and model
precision. Utilizing the one-class Support Vector Machine alongside a quantum
kernel, known for its classically challenging representational capacity,
notable improvements in average precision compared to classical counterparts
were observed in previous studies. Conventional calculations of these kernels,
however, present a quadratic time complexity concerning data size, posing
challenges in practical applications. To mitigate this, we explore two distinct
approaches: utilizing randomized measurements to evaluate the quantum kernel
and implementing the variable subsampling ensemble method, both targeting
linear time complexity. Experimental results demonstrate a substantial
reduction in training and inference times by up to 95\% and 25\% respectively,
employing these methods. Although unstable, the average precision of randomized
measurements discernibly surpasses that of the classical Radial Basis Function
kernel, suggesting a promising direction for further research in scalable,
efficient quantum computing applications in machine learning.

量子计算在机器学习任务中的潜力使得核计算和模型精度得到显著提高。通过使用一类支持向量机和量子核，相对于经典对照方法，先前研究观察到了显着的平均精度改进。然而，传统计算这些核的方法对于数据大小具有二次时间复杂度，这在实际应用中带来了挑战。为了缓解这个问题，我们探索了两种不同的方法：利用随机测量来评估量子核以及实施可变子采样集合方法，这两种方法都针对线性时间复杂度。实验结果表明，采用这些方法的训练和推断时间大大减少，分别达到了 95% 和 25%，同时随机测量的平均精度明显优于经典径向基函数核，为进一步研究可扩展、高效的量子计算在机器学习中的应用方向提供了有希望的线索。

高效量子异常检测：采用变量子采样和随机测量的单类支持向量机

Towards Efficient Quantum Anomaly Detection: One-Class SVMs using  Variable Subsampling and Randomized Measurements

In \cite{simon2023algorithms} we introduced four algorithms for the training
of neural support vector machines (NSVMs) and demonstrated their feasibility.
In this note we introduce neural quantum support vector machines, that is,
NSVMs with a quantum kernel, and extend our results to this setting.

我们引入了四种用于训练神经支持向量机的算法，并证明了它们的可行性。在本文章中，我们引入了神经量子支持向量机，即使用量子核的 NSVM，并将我们的结果扩展到该设置中。

神经量子支持向量机

On Neural Quantum Support Vector Machines

It has been hypothesized that quantum computers may lend themselves well to
applications in machine learning. In the present work, we analyze function
classes defined via quantum kernels. Quantum computers offer the possibility to
efficiently compute inner products of exponentially large density operators
that are classically hard to compute. However, having an exponentially large
feature space renders the problem of generalization hard. Furthermore, being
able to evaluate inner products in high dimensional spaces efficiently by
itself does not guarantee a quantum advantage, as already classically tractable
kernels can correspond to high- or infinite-dimensional reproducing kernel
Hilbert spaces (RKHS).
We analyze the spectral properties of quantum kernels and find that we can
expect an advantage if their RKHS is low dimensional and contains functions
that are hard to compute classically. If the target function is known to lie in
this class, this implies a quantum advantage, as the quantum computer can
encode this inductive bias, whereas there is no classically efficient way to
constrain the function class in the same way. However, we show that finding
suitable quantum kernels is not easy because the kernel evaluation might
require exponentially many measurements.
In conclusion, our message is a somewhat sobering one: we conjecture that
quantum machine learning models can offer speed-ups only if we manage to encode
knowledge about the problem at hand into quantum circuits, while encoding the
same bias into a classical model would be hard. These situations may plausibly
occur when learning on data generated by a quantum process, however, they
appear to be harder to come by for classical datasets.

通过分析量子核心函数类的光谱特性，我们发现如果其再现核希尔伯特空间具有低维度且包含传统计算难度的函数，则存在量子优势；然而，合适的量子核心函数不易找到，而该偏置相对于经典模型在量子机器学习模型速度提升方面是必须的。