Quantum Graphical Models (QGMs) generalize classical graphical models by
adopting the formalism for reasoning about uncertainty from quantum mechanics.
Unlike classical graphical models, QGMs represent uncertainty with density
matrices in complex Hilbert spaces. Hilbert space embeddings (HSEs) also
generalize Bayesian inference in Hilbert spaces. We investigate the link
between QGMs and HSEs and show that the sum rule and Bayes rule for QGMs are
equivalent to the kernel sum rule in HSEs and a special case of Nadaraya-Watson
kernel regression, respectively. We show that these operations can be
kernelized, and use these insights to propose a Hilbert Space Embedding of
Hidden Quantum Markov Models (HSE-HQMM) to model dynamics. We present
experimental results showing that HSE-HQMMs are competitive with
state-of-the-art models like LSTMs and PSRNNs on several datasets, while also
providing a nonparametric method for maintaining a probability distribution
over continuous-valued features.

研究提出了 HSE-HQMM 模型，通过量子图模型和 Hilbert 空间嵌入技术对物理过程进行建模，并在多个数据集上进行了比较试验。

希尔伯特空间中的学习与推断：量子图模型

Learning and Inference in Hilbert Space with Quantum Graphical Models

Hidden Quantum Markov Models (HQMMs) can be thought of as quantum
probabilistic graphical models that can model sequential data. We extend
previous work on HQMMs with three contributions: (1) we show how classical
hidden Markov models (HMMs) can be simulated on a quantum circuit, (2) we
reformulate HQMMs by relaxing the constraints for modeling HMMs on quantum
circuits, and (3) we present a learning algorithm to estimate the parameters of
an HQMM from data. While our algorithm requires further optimization to handle
larger datasets, we are able to evaluate our algorithm using several synthetic
datasets. We show that on HQMM generated data, our algorithm learns HQMMs with
the same number of hidden states and predictive accuracy as the true HQMMs,
while HMMs learned with the Baum-Welch algorithm require more states to match
the predictive accuracy.

本文提出了一种学习算法来从数据中估计 HQMM 的参数，该算法可以模拟经典的 HMM 并放宽了在量子电路上建模 HMM 的限制，同时在合成数据上的试验表明，我们的算法可以学习与真实 HQMM 相同数量状态和预测精度的 HQMM，而用 Baum-Welch 算法学习的 HMM 则需要更多状态才能匹配预测精度。