Explicit SoS lower bounds from high-dimensional expanders

TL;DR

This paper constructs explicit 3XOR instances based on high-dimensional expanders that are hard for a certain level of the Sum-of-Squares hierarchy, providing a deterministic alternative to previous random constructions.

Contribution

It introduces a deterministic polynomial-time construction of hard 3XOR instances using high-dimensional expanders, contrasting with prior random-based methods.

Findings

Explicit 3XOR instances are hard for O(√log n) levels of SOS hierarchy.

Construction is based on Lubotzky-Samuels-Vishne complexes (Ramanujan complexes).

Variables correspond to edges, not vertices, differing from previous work.

Abstract

We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author~(FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.