Efficiently stable presentations from error-correcting codes
Michael Chapman, Thomas Vidick, Henry Yuen

TL;DR
This paper introduces the concept of efficient stability for finite group presentations, linking it to error-correcting codes and quantum information theory, and demonstrates stable presentations with small size using Reed-Muller codes.
Contribution
It develops a new framework connecting stability of group presentations with testable error-correcting codes and quantum non-local games, providing explicit constructions with small size.
Findings
Stable presentations of ^k can be constructed from Reed-Muller codes.
Testability of codes correlates with weak stability of the associated presentations.
Explicit stable presentations of size polylogarithmic in k are achieved.
Abstract
We introduce a notion of \emph{efficient stability} for finite presentations of groups. Informally, a finite presentation using generators and relations is \emph{stable} if any map from to unitaries that approximately satisfies the relations (in the tracial norm) is close to the restriction of a representation of to the subset . This notion and variants thereof have been extensively studied in recent years, in part motivated by connections to property testing in computer science. The novelty in our work is the focus on \emph{efficiency}, which, informally, places an onus on small presentations -- in the sense of encoding length. The goal in this setup is to achieve non-trivial tradeoffs between the presentation length and its modulus of stability. With this goal in mind we analyze various natural examples of presentations. We provide a general method for…
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Taxonomy
TopicsComputability, Logic, AI Algorithms · semigroups and automata theory · Complexity and Algorithms in Graphs
