Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range

We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions f : {1,..., N} ! {1,..., M}, its polynomial degree is the same for all M N. Therefore, if we have a quantum query lower bound for some (possibly quite large) range M which is shown using the polynomials method, we immediately get the same lower bound for all ranges M N. In particular, we get Ω(N 1/3 ) and Ω(N 2/3 ) quantum lower bounds for collision and element distinctness with small range, respectively. As a corollary, we obtain a better lower bound on the polynomial degree of the two-level AND-OR tree.

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