Exponential Separation between Quantum Communication and Logarithm of Approximate Rank

Sinha, Makrand; Wolf, Ronald de

doi:10.1109/focs.2019.00062

Cited by 11 publications

(4 citation statements)

References 30 publications

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“…The 35-year-old conjecture is widely open today, inspiring numerous theorems and approaches even in the last few years [21][22][23][24][25][26][27].…”

Section: Conjecture 1 ([20]mentioning

confidence: 99%

A Note on the LogRank Conjecture in Communication Complexity

Grolmusz

2023

Mathematics

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The LogRank conjecture of Lovász and Saks (1988) is the most famous open problem in communication complexity theory. The statement is as follows: suppose that two players intend to compute a Boolean function f(x,y) when x is known for the first and y for the second player, and they may send and receive messages encoded with bits, then they can compute f(x,y) with exchanging (log rank(Mf))c bits, where Mf is a Boolean matrix, determined by function f. The problem is widely open and very popular, and it has resisted numerous attacks in the last 35 years. The best upper bound is still exponential in the bound of the conjecture. Unfortunately, we cannot prove the conjecture, but we present a communication protocol with (log rank(Mf))c bits, which computes a (somewhat) related quantity to f(x,y). The relation is characterized by the representation of low-degree, multi-linear polynomials modulo composite numbers. Our result may help to settle this long-open conjecture.

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“…The 35-year-old conjecture is widely open today, inspiring numerous theorems and approaches even in the last few years [21][22][23][24][25][26][27].…”

Section: Conjecture 1 ([20]mentioning

confidence: 99%

A Note on the LogRank Conjecture in Communication Complexity

Grolmusz

2023

Mathematics

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“…Sinha and de Wolf [SdW18] used the fooling distribution method, in independent and simultaneous work, to obtain the same Ω(m 1/3 ) lower bound on the quantum communication complexity of Sink • Xor. This differs from our techniques which we describe below.…”

Section: Independent Workmentioning

confidence: 99%

Quantum Log-Approximate-Rank Conjecture is Also False

Anshu¹,

Boddu²,

Touchette³

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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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.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Chattopadhyay,

Garg,

Sherif

2020

Preprint

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We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n 3 ) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [5] both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

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Exponential Separation between Quantum Communication and Logarithm of Approximate Rank

Cited by 11 publications

References 30 publications

A Note on the LogRank Conjecture in Communication Complexity

A Note on the LogRank Conjecture in Communication Complexity

Quantum Log-Approximate-Rank Conjecture is Also False

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

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