Quantum communication complexity of block-composed functions

A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the twoparty interactive model. The answer appears to be “No”. In 2002, Razborov proved this conjecture for so far the most general class of functions F (x, y) = fn(x1 · y1, x2 · y2, ..., xn · yn), where fn is a symmetric Boolean function on n Boolean inputs, and xi, yi are the i’th bit of x and y, respectively. His elegant proof critically depends on the symmetry of fn. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions F (x, y) is the “block-composition” of a “building block” gk : {0, 1}k × {0, 1}k → {0, 1}, and an fn : {0, 1}n → {0, 1}, such that F (x, y) = fn(gk(x1, y1), gk(x2, y2), ..., gk(xn, yn)), where xi and yi are the i’th k-bit block of x, y ∈ {0, 1}nk, respectively. We show that as long as gk itself is “hard” enough, its block-composition with an arbitrary fn has polynomially related quantum and classical communication complexities. Our approach gives an alternative proof for Razborov’s result (albeit with a slightly weaker parameter), and establishes new quantum lower bounds. For example, when gk is the Inner Product function with k = Ω(log n), the deterministic communication complexity of its block-composition with any fn is asymptotically at most the quantum complexity to the power of 7.