Stochastic processes are ubiquitous throughout the quantitative sciences and their study is of great interest across disciplinary boundaries. The complexity of stochastic process has been related to the minimal amount of memory about its past which one requires in order to predict its future . Recent studies have shown that quantum hidden Markov models can further reduce this memory requirement .
However, a precise understanding of the fundamental underpinnings of and resources required for this quantum advantage are not yet satisfactorily understood. In this talk we connect the best-known quantum models for stochastic process simulation to a powerful framework from condensed matter physics which is known as tensor network theory . More specifically, we establish a relationship between a stochastic process and a specific type of tensor network known as a matrix product state (MPS). The latter presents an efficient representation of quantum many-body states in condensed matter such as those of quantum spin chains.
Here, we construct a large quantum spin chain state, the measurement of which generates the stochastic process under consideration and present its MPS representation. MPS methods then allow us to establish a direct relationship between the amount of quantum memory required for process simulation and quantum correlations (i.e., quantum entanglement) in the spin chain representation. This relation sheds new light on the underlying benefits of quantum simulation models for stochastic processes by identifying quantum correlations as the direct analogue of a process’ complexity.
Lastly, we show that our approach provides an efficient way of constructing the best-known quantum models for stochastic simulation and we illustrate our framework with an example.