Quantum Tensor Networks, Stochastic Processes, and Weighted Automata
This work bridges gaps between physics, stochastic processes, and formal languages, potentially enabling cross-disciplinary application of methods, but it is incremental as it focuses on establishing theoretical equivalences rather than new practical advancements.
The paper tackles the problem of modeling joint probability distributions over sequences by showing equivalence between quantum tensor network models and models from stochastic processes and weighted automata in the limit of infinitely long sequences, connecting three previously separate research communities.
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.