Quantum Tensor Networks, Stochastic Processes, and Weighted Automata
Quantum Tensor Networks, Stochastic Processes, and Weighted Automata
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发表时间：
2020-10
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通讯作者：
Siddarth Srinivasan;Sandesh Adhikary;Jacob Miller;Guillaume Rabusseau;Byron Boots
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作者：
Siddarth Srinivasan;Sandesh Adhikary;Jacob Miller;Guillaume Rabusseau;Byron Boots
序列上的联合概率分布建模已经从许多角度进行了研究。物理学界开发了矩阵乘积状态，一种用于概率建模的张量-列车分解，其动机是需要对多体系统进行可追踪的建模。但是，在随机过程和加权自动机文献中也研究了类似的模型，很少研究这些工作如何相互关联。我们通过展示流行的量子张量网络模型的固定或统一版本如何在无限长序列的限制下在随机过程和加权自动机文献中具有等效表示来解决这一差距。我们展示了这三个社区中使用的模型之间的几个等效结果：（i）来自量子张量网络文献的矩阵乘积状态、玻恩机和局部纯化状态的一致变体，（ii）来自随机过程文献的预测状态表示、隐马尔可夫模型、范数可观测算子模型和隐量子马尔可夫模型，以及（iii）随机加权自动机，来自形式语言文献的概率自动机和二次自动机。这种联系可以为一个领域的成果和方法应用于另一个领域打开大门。
Modeling joint probability distributions over sequences has been studied from many perspectives. The physics community developed matrix product states, a tensor-train decomposition for probabilistic modeling, motivated by the need to tractably model many-body systems. But similar models have also been studied in the stochastic processes and weighted automata literature, with little work on how these bodies of work relate to each other. We address this gap by showing how stationary or uniform versions of popular quantum tensor network models have equivalent representations in the stochastic processes and weighted automata literature, in the limit of infinitely long sequences. We demonstrate several equivalence results between models used in these three communities: (i) uniform variants of matrix product states, Born machines and locally purified states from the quantum tensor networks literature, (ii) predictive state representations, hidden Markov models, norm-observable operator models and hidden quantum Markov models from the stochastic process literature,and (iii) stochastic weighted automata, probabilistic automata and quadratic automata from the formal languages literature. Such connections may open the door for results and methods developed in one area to be applied in another.