Quantum algorithms for hypergraph simplex finding

TL;DR

This paper develops quantum algorithms for hypergraph simplex finding, introducing new techniques like $ ext{alpha}$-symmetric learning graphs, and achieves improved query complexity for 4-simplex detection.

Contribution

It introduces $ ext{alpha}$-symmetric learning graphs and demonstrates their use in improving quantum algorithms for hypergraph simplex finding.

Findings

Quantum query algorithm for 4-simplex finding in rank-4 hypergraphs with $O(n^{2.46})$ complexity.

Rank-reduction property enables faster algorithms for lower-rank hypergraphs.

Conversion of nested Johnson graph quantum walks into adaptive learning graphs.

Abstract

We study the quantum query algorithms for simplex finding, a generalization of triangle finding to hypergraphs. This problem satisfies a rank-reduction property: a quantum query algorithm for finding simplices in rank-$r$ hypergraphs can be turned into a faster algorithm for finding simplices in rank-$(r-1)$ hypergraphs. We then show that every nested Johnson graph quantum walk (with any constant number of nested levels) can be converted into an adaptive learning graph. Then, we introduce the concept of $\alpha$-symmetric learning graphs, which is a useful framework for designing and analyzing complex quantum search algorithms. Inspired by the work of Le Gall, Nishimura, and Tani (2016) on $3$-simplex finding, we use our new technique to obtain an algorithm for $4$-simplex finding in rank-$4$ hypergraphs with $O(n^{2.46})$ quantum query cost, improving the trivial $O(n^{2.5})$ algorithm.