Statistical mechanics and hierarchy of learnability in quantum codes: From generalized symmetries to non-Abelian topological orders

Abstract

Local weak measurements provide noisy information about observables encoded in a quantum code. We formulate the corresponding Bayesian inference problem for a broad class of topological stabilizer codes by mapping it to classical spin models with generalized symmetries. When the wavefunction deformation and Gaussian measurement strengths coincide, the replicated model acquires an enlarged symmetry. We use this symmetry to derive exact identities among correlators averaged over measurement records that quantify the learnability of nonlocal observables associated with fractionalized excitations. Together with correlation inequalities in the associated clean ferromagnetic models, these identities establish hierarchies of learnability: if two observables remain learnable at long distances, their product must also remain learnable. We demonstrate these relations in the rank-2 toric code and the X-cube model, whose classical descriptions exhibit modulated and subsystem symmetries, respectively. We further extend the analysis to finite group quantum double models, including non-Abelian ones. We derive exact relations between learnability and correlations in the corresponding classical models, and show that the representation structure and fusion rules of anyons impose further constraints under suitable positivity conditions. These results constrain which types of nonlocal information can be learned simultaneously, providing a guide to learning phase diagrams in quantum codes under local weak measurements.