Optimal parallel quantum query algorithms

We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. We show tight bounds for a number of problems, specifically Θ((n/p)^2/3) p-parallel queries for element distinctness and Θ((n/p)^{k/(k + 1)}) for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to f’s block sensitivity.