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)logn), 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(logn) 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 clog∗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
NORn∘∧) is O(Q(NORn)).
Perhaps somewhat surprisingly, we show that when ∙=⊕, then
the extra logn factor in the BCW simulation is unavoidable. In other words,
we exhibit a total function F:{−1,1}n→{−1,1} such that Qcc(F∘⊕)=Θ(Q(F)logn).
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 Qcc(F∘∙)=ω(Q(F))