Explicit Lossless Vertex Expanders

2025-04-21

AI Summary

This work constructs, for the first time explicitly, constant-degree lossless vertex expanders: an infinite family of $d$-regular graphs, for any $varepsilon > 0$ and sufficiently large $d$, such that every small vertex subset $S$ has at least $(1-varepsilon)d|S|$ distinct neighbors—i.e., approximately $(1-2varepsilon)d|S|$ unique neighbors. Method: We introduce a graph product framework combining small-scale lossless expanders with Ramanujan Cayley cubical complexes, leveraging their underlying free group action structure. The construction naturally extends to arbitrarily imbalanced biregular bipartite graphs. Contribution/Results: This yields the first explicit, constant-degree, strictly lossless vertex expander family—achieving combinatorial optimality (near-optimal expansion) and efficient decodability. As a corollary, we obtain a new class of quantum low-density parity-check (LDPC) codes decodable in linear time. The result establishes a novel paradigm bridging high-dimensional expanders and quantum error correction, resolving a long-standing open problem in explicit expander construction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $varepsilon>0$ 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 $(1-varepsilon)d|S|$ neighbors (which implies $(1-2varepsilon)d|S|$ 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.

Problem

• Construct explicit constant-degree lossless vertex expanders
• Extend to biregular bipartite graphs with strong expansion
• Realize new families of quantum LDPC codes

Innovation

• Explicit constant-degree lossless vertex expanders
• Product of expander with Ramanujan Cayley complexes
• Free group action enabling linear-time decoding