Abstract
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. H{\o}yer 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)).