Quantum Query-to-Communication Simulation Needs a Logarithmic Overhead
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Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function f : {-1, 1}^n →{-1, 1} and ∙ : {-1, 1}^2 →{-1, 1} the two-party bounded-error quantum communication complexity of (f ∘∙) is O(Q(f) log n), where Q(f) is the bounded-error quantum query complexity of f. Note that the bounded-error randomized communication complexity of (f ∘∙) is bounded by O(R(f)), where R(f) denotes the bounded-error randomized query complexity of f. Thus, the BCW simulation has an extra O(log n) factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Høyer and de Wolf (STACS'02) showed that for the Set-Disjointness function, this can be reduced to c^log^* n for some constant c, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is NOR_n ∘∧) is O(Q(NOR_n)). Perhaps somewhat surprisingly, we show that when ∙ = ⊕, then the extra log n factor in the BCW simulation is unavoidable. In other words, we exhibit a total function F : {-1, 1}^n →{-1, 1} such that Q^cc(F ∘⊕) = Θ(Q(F) log n). To the best of our knowledge, it was not even known prior to this work whether there existed a total function F and 2-bit function ∙, such that Q^cc(F ∘∙) = ω(Q(F)).
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