Title: Quantum Log-Approximate-Rank Conjecture is also False
|Affiliation:||Institute for Quantum Computing - University of Waterloo|
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 `f', hence refuting the log approximate rank conjecture of Lee and Shraibman . 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 `f'. 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 . Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.
Joint work with Naresh Good Boddu and Dave Touchette (https://arxiv.org/abs/1811.10525)
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