Fault-Tolerant Logical Gates In Holographic Stabilizer Codes Are Severely Restricted

We evaluate the usefulness of holographic stabilizer codes for fault-tolerant quantum computing 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, and certain codes whose logical algebras have nontrivial 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 an entanglement wedge map for any subsystem code, and a thorough classification of different correctability properties for regions in a subsystem code.