Quantum Log-Approximate-Rank Conjecture is also False
Abstract
In a recent breakthrough result, Chattopadhyay, Mande and Sherif [ECCC TR18-17] showed an exponential separation between the log approximate rank and randomized communication complexity of a total function , hence refuting the log approximate rank conjecture of Lee and Shraibman [2009]. We provide an alternate proof of their randomized communication complexity lower bound using the information complexity approach. Using the intuition developed there, we derive a polynomially-related quantum communication complexity lower bound using the quantum information complexity approach, thus providing an exponential separation between the log approximate rank and quantum communication complexity of . Previously, the best known separation between these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain, Kothari and Lee [CCC, 2017]. This settles one of the main question left open by Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank conjecture of Lee and Shraibman [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.
Cite
@article{arxiv.1811.10525,
title = {Quantum Log-Approximate-Rank Conjecture is also False},
author = {Anurag Anshu and Naresh Goud Boddu and Dave Touchette},
journal= {arXiv preprint arXiv:1811.10525},
year = {2020}
}
Comments
21 pages. The same lower bound has been obtained independently and simultaneously by Makrand Sinha and Ronald de Wolf. Part of the preliminaries taken from arXiv:1611.05754