When are approximate unitary representations of a group G close to exact representations? The answer depends on the group G and on the notion of closeness. Results about group stability typically do not depend on the specific presentation of the group G: bounds on the stability for one presentation of G can be generically translated to bounds for any other presentation -- with some blow up that depends on the presentations. However, many computer science and combinatorics applications focus on groups G with efficient presentations, where the number of generators and relations is small compared to the group size. In this setting, one would like to obtain good tradeoffs between the presentation size and the closeness of approximate representations to exact representations (we call this the modulus of stability).

We prove that the group $\mathbb{Z}_2^k$ has a presentation that is both stable and efficient: the presentation size and the modulus of stability scale favorably with the group size. The presentation is constructed using well-studied error-correcting codes in theoretical computer science and information theory. The stability and efficiency of this presentation underlies one of the key components of the proof of MIP* = RE and the corresponding resolution of Connes' embedding problem.

Bio: Henry Yuen is an Assistant Professor of Computer Science at Columbia University. His research focuses on the interplay between quantum computing, complexity theory, cryptography, and information theory. Yuen received a BA in mathematics from the University of Southern California in 2010, and received his PhD in computer science at MIT in 2016. He is a recipient of the NSF CAREER award and a Sloan Fellowship.