Higher Abelian Quantum Double Models

2025/09/16 by Flores, Jorge Acuña, De Nittis, Giuseppe, Espiro, Javier Lorca
#47L40 #82B20 #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 81R15 #Secondary:46L30

paper · doi:10.48550/arxiv.2509.12864

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 FG_QDM(\mathfrakA). A central result is the construction of the algebra of logical operators \mathfrakA_log := \mathfrakK'/\mathfrakJ, where the redundancy ideal \mathfrakJ quotients out operators that act trivially on the ground state space. We prove a homeomorphism between the state space of \mathfrakA_log and FG_QDM(\mathfrakA), 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, \mathfrakA_log is isomorphic to C(X_c) ⊗ B(\mathfrakh_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.

Citations

• Topological phases of non-interacting systems: A general approach based on states
• Classification of the anyon sectors of Kitaev's quantum double model
• A dualization approach to the Ground State Subspace Classification of Abelian Higher Gauge Symmetry Models
• Topological Order from a Cohomological and Higher Gauge Theory perspective
• Quantum Computation with Topological Codes: from qubit to topological fault-tolerance
• Fault-tolerant quantum computation by anyons

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