Lower bound for quantum phase estimation

We obtain a query lower bound for quantum algorithms solving the phase estimation problem. Our analysis generalizes existing lower-bound approaches to the case where the oracle Q is given by controlled powers Q(p) of Q, as it is, for example, in Shor's order-finding algorithm. In this setting we will prove a Omega(log 1/epsilon) lower bound for the number of applications of Q(1)(p), Q(2)(p),.... This bound is tight due to a matching upper bound. We obtain the lower bound using a technique based on frequency analysis.