Causal Architecture in Hidden Quantum Markov Models

Abstract

We introduce a class of causal hidden quantum Markov models (cHQMMs) that reverse the usual order of hidden updates and emissions compared to conventional HQMMs. Using a simple qubit model with a rotating hidden state and sharp measurements, we show that these two architectures-emission then transition versus transition then emission-generally produce different quantum processes. They can be distinguished by measurements at arbitrarily late times, no matter how the hidden system is initialized, and even when the two models start from different initial states. This means that the two orders of operations lead to genuinely different observable behaviors that cannot be reconciled by waiting longer or by choosing special initial conditions. At the same time, we prove that the two architectures become equivalent when they arise from entangled liftings of classical hidden Markov models, sharing the same classical statistics. This identifies a clear dividing line between classical and genuinely quantum hidden memory. Our findings highlight causal HQMMs as a useful tool for studying and distinguishing quantum memory effects in sequential processes.