Lifting randomized query complexity to randomized communication complexity

We show that for any (partial) query function f : {0, 1} → {0, 1}, the randomized communication complexity of f composed with Indexm (with m = poly(n)) is at least the randomized query complexity of f times log n. Here Indexm : [m] × {0, 1} → {0, 1} is defined as Indexm(x, y) = yx (the xth bit of y). Our proof follows on the lines of Raz and Mckenzie [RM99] (and its generalization due to [GPW15]), who showed a lifting theorem for deterministic query complexity to deterministic communication complexity. Our proof deviates from theirs in an important fashion that we consider partitions of rectangles into many sub-rectangles, as opposed to a particular sub-rectangle with desirable properties, as considered by Raz and McKenzie. As a consequence of our main result, some known separations between quantum and classical communication complexities follow from analogous separations in the query world.