Quantum Query-to-Communication Simulation Needs a Logarithmic Overhead

Abstract

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function $f:\{-1,1{\}}^n\to \{-1,1\}$ and $\bullet :\{-1,1{\}}^2\to \{-1,1\}$ the two-party bounded-error quantum communication complexity of $(f\circ \bullet )$ 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 $(f\circ \bullet )$ 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 $c^{{\log }^{\ast }n}$ 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 ${\mathsf{NOR}}_n\circ \wedge$) is $O(Q({\mathsf{NOR}}_n))$. Perhaps somewhat surprisingly, we show that when $\bullet =\oplus$, then the extra $\log n$ factor in the BCW simulation is unavoidable. In other words, we exhibit a total function $F:\{-1,1{\}}^n\to \{-1,1\}$ such that $Q^{cc}(F\circ \oplus )=\Theta (Q(F)\log n)$. 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 $\bullet$, such that $Q^{cc}(F\circ \bullet )=\omega (Q(F))$.