On statistical query sampling and NMR quantum computing

We introduce a "statistical query sampling" model, in which the goal of an algorithm is to produce an element in a hidden set S⊆{0,1}ⁿ with reasonable probability. The algorithm gains information about S through oracle calls (statistical queries), where the algorithm submits a query function g(·) and receives an approximation to Pr_x∈S[g(x)=1]. We show how this model is related to NMR quantum computing, in which only statistical properties of an ensemble of quantum systems can be measured, and in particular to the question of whether one can translate standard quantum algorithms to the NMR setting without putting all of their classical postprocessing into the quantum system. Using Fourier analysis techniques developed in the related context of statistical query learning, we prove a number of lower bounds (both information-theoretic and cryptographic) on the ability of algorithms to produce an x∈S, even when the set S is fairly simple. These lower bounds point out a difficulty in efficiently applying NMR quantum computing to algorithms such as Shor\´s and Simon\´s algorithm that involve significant classical postprocessing. We also explicitly relate the notion of statistical query sampling to that of statistical query learning.