Gauge Structures: From Stabilizer Codes to Continuum Models

Abstract

Stabilizer codes are a powerful method for implementing fault-tolerant quantum memory and in the case of topological codes, they form useful models for topological phases of matter. In this paper, we discuss the theory of stabilizer codes as a discrete version of a linear gauge structure, a concept we introduce here. A linear gauge structure captures all the familiar ingredients of $U(1)$ gauge theory including a generalization of charge conservation, Maxwell equations and topological terms. Using this connection, we prove some important results for stabilizer codes which can be used to characterize their error-correction properties. However, this perspective does not depend on any particular Hamiltonian or Lagrangian, which as a consequence is agnostic to the source of the gauge redundancy. Based upon the connection to stabilizer codes, we view the source of the gauge redundancy as an ambiguity in the tensor product structure of the Hilbert space. That is, the gauge provides a set of equivalent but arbitrary ways to factorize the Hilbert space. We apply this formalism to the $d=2$ and $3$ toric code as well as the paradigm fracton models, X-Cube and Haah's cubic code. From this perspective, we are able to map all these models to some continuum version which captures all or some of the fractonic and topological features of their parent discrete models.

arXiv comment27 pages, 6 figures, removed a section, refined proof of Thm. 3, added connection to gauging and ungauge procedure

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• Fracton fusion and statistics
• Emergent fractons and algebraic quantum liquid from plaquette melting transitions
• Gauging permutation symmetries as a route to non-Abelian fractons
• Fractons from polarons
• Quantum robustness of fracton phases
• Hyperbolic Fracton Model, Subsystem Symmetry, and Holography II: The Dual Eight-Vertex Model
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• Electric Circuit Realizations of Fracton Physics
• Asymptotic symmetries and soft charges of fractons