Higher Abelian Quantum Double Models

This paper focuses on the generalized version of the quantum double model on arbitrary $N$-dimensional simplicial complexes with finite local regularity. The core of our analysis is a detailed characterization of the frustration-free ground state space ${\mathrm{F}\mathrm{G}}_{\mathrm{Q}\mathrm{D}\mathrm{M}}(\mathfrak{A})$. A central result is the construction of the algebra of logical operators ${\mathfrak{A}}_{\mathrm{l}\mathrm{o}\mathrm{g}}:={\mathfrak{K}}^{\mathrm{\prime }}\mathrm{/}\mathfrak{J}$, where the redundancy ideal $\mathfrak{J}$ quotients out operators that act trivially on the ground state space. We prove a homeomorphism between the state space of ${\mathfrak{A}}_{\mathrm{l}\mathrm{o}\mathrm{g}}$ and ${\mathrm{F}\mathrm{G}}_{\mathrm{Q}\mathrm{D}\mathrm{M}}(\mathfrak{A})$, effectively classifying all frustration-free ground states. This logical algebra is shown to exhibit generalized Canonical Commutation Relations (CCR). When the relevant (co)homology groups are finite, ${\mathfrak{A}}_{\mathrm{l}\mathrm{o}\mathrm{g}}$ is isomorphic to $C(X_c)\otimes \mathcal{B}({\mathfrak{h}}_q)$, revealing that the ground state space can encode $c$ classical bits and $q$ quantum bits (qubits), providing a precise measure of its information storage capacity.