Cocycle Actions on Hidden Quantum Markov Models: Symmetry Protection and Topological Order

Abstract: 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.

• The paper presents an operator-algebraic framework for HQMMs that rigorously encodes quantum stochastic dynamics via cocycle actions.
• It demonstrates symmetry protection in one-dimensional quantum spin systems by intertwining projective representations in the hidden layer with linear ones in the observable layer, validated on the AKLT chain.
• The findings offer implications for robust quantum memory and machine learning through the preservation and classification of SPT phases using group cohomology.

Cocycle Actions on Hidden Quantum Markov Models: Symmetry Protection and Topological Order

Algebraic Framework and Symmetry Structure of HQMMs

The paper introduces a comprehensive operator-algebraic framework for symmetry actions on Hidden Quantum Markov Models (HQMMs), with particular emphasis on applications to one-dimensional quantum spin systems and the representation of symmetry-protected topological (SPT) phases. HQMMs extend classical hidden Markov models by incorporating quantum degrees of freedom both in the observed and latent layers. The formalism leverages separable Hilbert spaces and $C^*$-algebraic structures to rigorously encode quantum stochastic dynamics, including generative triples $(\phi_0, \mathcal{E}_H, \mathcal{E}_{H,O})$, where $\phi_0$ is the initial state, $\mathcal{E}_H$ is the hidden transition map, and $\mathcal{E}_{H,O}$ is the emission map.

A salient feature is the projective action of a symmetry group $G$ on the hidden sector, parameterized by a $2$-cocycle $\omega \in H^2(G, \mathrm{U}(1))$, while the observable layer transforms under a conventional linear unitary representation. The symmetry action is orchestrated through three fundamental conditions: invariance of the initial state, equivariance of the hidden transition, and covariance of the emission map. The emission map serves as a cocycle intertwiner, mediating between the projective hidden dynamics and linear observable dynamics, thus absorbing the projective anomaly at the interface and enforcing global invariance of the HQMM state under the combined symmetry action.

Group Cohomology and Classification of SPT Phases

The connection between HQMMs and SPT phases is formalized by identifying the projective representation class $[\omega]$ as the topological invariant, echoing the established group cohomological classification of one-dimensional bosonic SPT phases. The composition laws and cohomological constraints on tensor product representations are analyzed in detail. The paper rigorously proves—via operator-algebraic constructions and inductive limit arguments—that the global HQMM state remains invariant under $G$ for both conventional (measurement-then-evolution) and causal (evolution-then-measurement) architectures. The covariance of the emission map is explicitly characterized as trivializing the cocycle obstruction. This algebraic invariance theorem encapsulates the robust preservation of SPT order within HQMMs.

Application to the AKLT Chain

The framework is explicitly instantiated in the paradigmatic AKLT chain, whose ground state is a well-known matrix product state representing a nontrivial SPT phase protected by $(\phi_0, \mathcal{E}_H, \mathcal{E}_{H,O})$0 symmetry. The hidden virtual space carries a nontrivial projective spin-$(\phi_0, \mathcal{E}_H, \mathcal{E}_{H,O})$1 representation with cocycle $(\phi_0, \mathcal{E}_H, \mathcal{E}_{H,O})$2, while the observable layer transforms linearly (spin-1). The emission map is realized via the AKLT tensors, enforcing the intertwining condition that symmetry actions on physical indices correspond to projective conjugation on virtual tensors.

The paper demonstrates that the HQMM formalism reproduces the AKLT state's SPT properties and confirms global $(\phi_0, \mathcal{E}_H, \mathcal{E}_{H,O})$3 invariance for both causal structures. The uniqueness of the $(\phi_0, \mathcal{E}_H, \mathcal{E}_{H,O})$4-invariant initial state (maximal mixed state) is established by Schur's lemma, and the emission map's covariance is shown to exactly absorb the projective anomaly, certifying that the topological order persists under quantum Markovian evolution.

Theoretical and Practical Implications

The operator-algebraic HQMM framework provides a rigorous bridge between quantum stochastic processes, tensor-network representations (including matrix product states), and the mathematical structure of topological phases. The algebraic symmetry-protection conditions are applicable beyond the AKLT chain, furnishing a classifying principle for HQMMs associated with arbitrary cocycles and symmetry groups.

The approach yields practical implications for quantum information protocols and quantum machine learning: HQMMs model sequential quantum data with latent memory, and symmetry-protected architectures offer robust quantum state preparation and learning algorithms invariant under physical symmetries. The cocycle intertwiner principle suggests robust quantum memory models where SPT order persists under local Markovian noise.

Theoretical extensions include the classification of HQMMs by higher group cohomology for multidimensional systems, analysis of twisted equivariant structures, and formulation of numerical SPT invariants. The paper positions the algebraic framework as a foundation for further research in ergodicity, mixing behavior, quantum memory, and symmetry-aware learning.

Conclusion

This paper rigorously establishes HQMMs as a natural and mathematically robust framework for encoding and classifying symmetry-protected topological order in quantum stochastic systems. By leveraging group cohomology and projective representation theory, it unifies tensor network structures and quantum Markov processes, offering both theoretical depth and practical utility. The identification of emission maps as cocycle intertwiners opens new vistas for quantum memory, learning, and classification of topological phases within operator-algebraic dynamics.