Causal Architecture in Hidden Quantum Markov Models

Abstract: We introduce a class of causal hidden quantum Markov models (cHQMMs) that refine standard HQMMs by explicitly reversing the order between hidden updates and emissions. Through a minimal qubit model, we show that the conventional "emission-then-transition" and the alternative "transition-then-emission" architectures generally generate nonequivalent quantum processes, with distinct temporal correlation structures and different patterns of entanglement across time. At the same time, we prove that these two classes coincide for entangled lifting of classical hidden Markov models, where they share the same classical reduced process, thereby identifying a sharp boundary between classical and genuinely quantum hidden memory. These features suggest potential utility for modeling and analyzing quantum memory in sequential quantum processes.