Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes

TL;DR

This paper explores the differences between classical and quantum models of stochastic processes, highlighting the existence of processes where quantum models outperform classical ones and emphasizing the importance of choosing appropriate quantum memory measures.

Contribution

It reveals that some processes lack a strongly minimal quantum model, unlike classical epsilon-machines, and shows quantum memory optimization depends on the chosen measure.

Findings

Classical epsilon-machines are strongly minimal for Rènyi-based measures.

Certain processes have no strongly minimal pure-state quantum models.

Quantum memory optimization varies with the selected measure.

Abstract

Among the predictive hidden Markov models that describe a given stochastic process, the {\epsilon}-machine is strongly minimal in that it minimizes every R\'enyi-based memory measure. Quantum models can be smaller still. In contrast with the {\epsilon}-machine's unique role in the classical setting, however, among the class of processes described by pure-state hidden quantum Markov models, there are those for which there does not exist any strongly minimal model. Quantum memory optimization then depends on which memory measure best matches a given problem circumstance.