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.
Highlights
The anti-de Sitter/conformal field theory (AdS/CFT) correspondence [1,2] is a fruitful instance of the holographic principle [3,4] from high-energy theory—it refers to a duality between a theory of quantum gravity in D + 1 dimensions and a conformally invariant quantum field theory on the D-dimensional boundary of the original space
We show that for sufficiently localitypreserving physical gates, the restricted logical action on a sufficiently small region of the bulk cannot implement a non-Clifford gate for any stabilizer code with a holographic structure that captures certain properties of AdS/CFT in any spatial dimension
We introduce a general definition of entanglement wedge, termed the “maximal entanglement wedge,” which is defined for an arbitrary subsystem code without any assumptions of additional structure such as a particular geometry
Summary
The anti-de Sitter/conformal field theory (AdS/CFT) correspondence [1,2] is a fruitful instance of the holographic principle [3,4] from high-energy theory—it refers to a duality between a theory of quantum gravity in D + 1 dimensions and a conformally invariant quantum field theory on the D-dimensional boundary of the original space. Topological codes possess additional geometric structure, which results in a wider class of fault-tolerant gates that subsumes the set of transversal gates: those implementable by locality-preserving operations. It has been shown that locality-preserving gates in topological stabilizer subsystem codes [28,31,32] in D spatial dimensions can implement only gates from CD, the Dth level of the Clifford hierarchy, which is a series of increasing sets of gates that includes the Pauli group (C1) and the Clifford group (C2). We show that the restrictions are much severer than this—namely, that locality-preserving gates for such holographic codes are dimension-independently restricted to the Clifford group, C2
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