  This article presents a quantum computing approach to designing of similarity
measures and kernels for classification of stochastic symbolic time series. In
the area of machine learning, kernels are important components of various
similarity-based classification, clustering, and regression algorithms. An
effective strategy for devising problem-specific kernels is leveraging existing
generative models of the example space. In this study we assume that a quantum
generative model, known as quantum hidden Markov model (QHMM), describes the
underlying distributions of the examples. The sequence structure and
probability are determined by transitions within model's density operator
space. Consequently, the QHMM defines a mapping from the example space into the
broader quantum space of density operators. Sequence similarity is evaluated
using divergence measures such as trace and Bures distances between quantum
states. We conducted extensive simulations to explore the relationship between
the distribution of kernel-estimated similarity and the dimensionality of the
QHMMs Hilbert space. As anticipated, a higher dimension of the Hilbert space
corresponds to greater sequence distances and a more distinct separation of the
examples. To empirically evaluate the performance of the kernels, we defined
classification tasks based on a simplified generative model of directional
price movement in the stock market. We implemented two widely-used kernel-based
algorithms - support vector machines and k-nearest neighbors - using both
classical and quantum kernels. Across all classification task scenarios, the
quantum kernels consistently demonstrated superior performance compared to
their classical counterparts.
