The Garden Hose Complexity for the Equality Function

TL;DR

This paper investigates the garden hose complexity of the equality function, introducing new bounds, symmetries, and the concept of garden hose permutation groups to advance understanding in quantum communication complexity.

Contribution

It improves existing bounds on garden hose complexity for the equality function using a novel approach and introduces the concept of garden hose permutation groups.

Findings

Improved bounds on garden hose complexity for equality function

Discovery of symmetries leading to garden hose permutation groups

Development of a new simulated annealing based solver

Abstract

The garden hose complexity is a new communication complexity introduced by H. Buhrman, S. Fehr, C. Schaffner and F. Speelman [BFSS13] to analyze position-based cryptography protocols in the quantum setting. We focus on the garden hose complexity of the equality function, and improve on the bounds of O. Margalit and A. Matsliah[MM12] with the help of a new approach and of our handmade simulated annealing based solver. We have also found beautiful symmetries of the solutions that have lead us to develop the notion of garden hose permutation groups. Then, exploiting this new concept, we get even further, although several interesting open problems remain.