Quantum Log-Approximate-Rank Conjecture is Also False

Abstract

In a recent breakthrough result, Chattopadhyay, Mande and Sherif 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 [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 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 [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

“…Thus, conditionally, we get the following improvements over the results in [5]: (1) We narrow the gap between approximate rank and randomized communication complexity from quartic to cubic. (2) We expand the gap between log-approximate-rank and randomized complexity from O(log n) vs. √ n to O(log n) vs. n, thus yielding essentially the strongest possible refutation of the LARC, under plausible assumptions.…”

“…If we have to find a total function with an exponential gap between the quantum communication and randomized communication complexities (if one exists at all), then the function should also have an exponential separation between log of approximate rank and randomized communication complexity. 1 However, it was shown by Anshu et al [1] and Sinha and de Wolf [23] that the function of [5] has large quantum communication complexity (hence refuting the quantum version of LARC as well). This motivates the search for other examples refuting the LARC.…”

“…In other words, a communication protocol cannot be more efficient than naively simulating the optimal RPDT. The strongest evidence for such an assertion is the result of Hatami, Hosseini and Lovett [16] who show that if f has deterministic PDT cost c, then f • XOR has deterministic communication complexity c Ω (1) . While no general result exists for the randomized model, the community believes it to be plausible.…”

Exponential Separation between Quantum Communication and Logarithm of Approximate Rank

“…The main question left open by this work, as well as by [CMS18,ABT18], is of course the status of the (non-approximate) log-rank conjecture itself. The proof that the sink function has low approximate rank crucially uses the fact that the identity matrix has low approximate rank (which follows from the fact that the equality function has low randomized communication complexity).…”

“…Independent Work. In independent and simultaneous work, Anshu, Boddu and Touchette [ABT18] obtained the same Ω(t 1/3 ) lower bound using a reduction to quantum information complexity of the equality function, but our techniques to prove Theorem 1.2 are different, as we describe below.…”