Abstract: We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any ε>0 and sufficiently large d, we give an explicit construction of an infinite family of d-regular graphs where every small set S of vertices has (1−ε)d|S| neighbors (which implies (1−2ε)d|S| unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm. Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.
Based on joint work with Jun-Ting Hsieh, Alexander Lubotzky, Assaf Reiner, and Rachel Yun Zhang (https://arxiv.org/abs/2504.15087)
Bio: Sidhanth is an Assistant Professor in the Computer Science department at Northwestern University. He is broadly interested in theoretical computer science and probability theory, and his specific recent interests include mixing times of Markov chains, high-dimensional expansion, random matrix theory, and error-correcting codes.