Quantum Lower Bounds by Polynomials

We examine the number T of oracle calls that a quantum network requires to compute some Boolean function on {0,1}^N in the so-called black-box model, where the input is given as an oracle. We show that the acceptance probability of a network can be written as an N-variate polynomial of the input, having degree at most 2T. Using lower bounds on the degrees of polynomials that equal or approximate Boolean functions, we derive lower bounds on T. We give precise (up to a constant multiplicative factor) characterizations of T for all symmetric f, both if we require exact computation and if we allow some error probability. More precisely, to compute PARITY, T=N/2 is necessary and sufficient in both settings, while for OR and AND we need N queries for exact computation and Theta(sqrt{N}) for bounded-error computation. If we restrict to inputs that satisfy some promise, then quantum networks can sometimes achieve an exponential speed-up compared to classical randomized methods (Deutsch-Jozsa, Simon). However, we prove that without promises quantum computing can achieve only a limited speed-up in the black-box model: if a quantum network can compute some total Boolean function f using T oracle calls with bounded error, then a classical deterministic algorithm can compute f exactly in O(T^6) oracle calls