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We evaluate the usefulness of holographic stabilizer codes for practical purposes by studying their allowed sets of fault-tolerantly implementable gates. We treat them as subsystem codes and show that the set of transversally implementable logical operations is contained in the Clifford group for sufficiently localized logical subsystems. As well as proving this concretely for several specific codes, we argue that this restriction naturally arises in any stabilizer subsystem code that comes close to capturing certain properties of holography. We extend these results to approximate encodings, locality-preserving gates, certain codes whose logical algebras have non-trivial centers, and discuss cases where restrictions can be made to other levels of the Clifford hierarchy. A few auxiliary results may also be of interest, including a general definition of entanglement wedge map for any subsystem code, and a thorough classification of different correctability properties for regions in a subsystem code. CONTENTS1 This does not include the more general class of gates and codes with boundaries considered in Refs. [28,29].

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In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group G f of the ground state splits aswhere Z f 2 is generated by fermion parity, (−1) F . In contrast, (2+1)D topological superconductors (TSC) can host unpaired MZMs at defects even when G f is not of the form G b × Z f 2 . In this paper we study how G f together with the chiral central charge c− strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a non-perturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as U(1) f H, SU(2) f × H, U(2) f H, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr2RuO4, transition metal dichalcogenides and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models. CONTENTS 2 D. MZMs in reflection symmetric systems 1. TSCs with mirror symmetry