Cocycle Actions on Hidden Quantum Markov Models: Symmetry Protection and Topological Order
We develop a symmetry action framework for hidden quantum Markov models (HQMMs) tailored to one-dimensional quantum spin systems and symmetry-protected topological (SPT) phases. In our setting, a symmetry group $G$ acts projectively on the hidden (virtual) degrees of freedom and linearly on the physical observation space, yielding a global HQMM state that is invariant under the combined action of $G$ for both conventional and causal (input--output) structures. We show that such symmetry actions are naturally classified by a group-cohomology $2$-cocycle $[ω] \in H^{2}(G,\mathrm{U}(1))$, in direct analogy with the standard cohomological classification of one-dimensional bosonic SPT phases via projective edge representations. As an explicit example, we apply this construction to the Affleck--Kennedy--Lieb--Tasaki (AKLT) chain, where the hidden layer carries a nontrivial class $[ω] \in H^{2}(\mathrm{SO}(3),\mathrm{U}(1))$ encoding its SPT order. In this case the HQMM formalism reproduces the known SPT properties of the AKLT state while providing a stochastic, Markovian description of the underlying virtual dynamics. Our results establish HQMMs as a natural bridge between quantum stochastic processes, tensor-network descriptions of many-body systems, and symmetry-protected topological order.