Higher Abelian Quantum Double Models

Abstract

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 $\mathrm{FG}_{\mathrm{QDM}}(\mathfrak{A})$. A central result is the construction of the algebra of logical operators $\mathfrak{A}_{\mathrm{log}} := \mathfrak{K}'/\mathfrak{J}$, 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 $\mathfrak{A}_{\mathrm{log}}$ and $\mathrm{FG}_{\mathrm{QDM}}(\mathfrak{A})$, 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, $\mathfrak{A}_{\mathrm{log}}$ 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.