Fault-tolerant logical gates in holographic stabilizer codes are severely restricted

TL;DR

This paper investigates the limitations of fault-tolerant logical gates in holographic stabilizer codes, showing they are largely restricted to the Clifford group, which impacts their practical utility.

Contribution

It proves that in holographic stabilizer codes, fault-tolerant gates are confined to the Clifford group and extends this restriction to various code types and approximate encodings.

Findings

Transversal logical gates are within the Clifford group for localized subsystems.

Restrictions on logical gates are inherent in holographic stabilizer codes.

Extensions to approximate and non-trivial center codes are discussed.

Abstract

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.