Matrix Product States for Quantum Stochastic Modelling

The pursuit of simplicity underlies most of quantitative science. In stochastic modeling, there has been significant effort towards finding models that predict a process' future using minimal information from its past. Meanwhile, in condensed matter physics, finding efficient representations for large quantum many-body systems is a topic of critical concern -- exemplified by the development of the matrix product state (MPS) formalism. In this letter, we connect these two distinct fields. Specifically, we associate each stochastic process with a suitable quantum state of a spin-chain. We show that the optimal predictive model for the process leads directly to the MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of q-simulators -- the currently best known predictive quantum models for stochastic processes. We show that the memory requirements of the these models directly coincide with the bipartite entanglement of an associated spin-chain, providing an analytical connection between quantum correlations in many body physics and the complexity of stochastic modeling.