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Quantum algorithms for the Goldreich-Levin learning problem

The Goldreich-Levin algorithm was originally proposed for a cryptographic purpose and then applied to learning. The algorithm is to find some larger Walsh coefficients of an $n$ variable Boolean function. Roughly speaking, it takes a $poly(n,\frac{1}ε\log\frac{1}δ)$ time to output the vectors $w$ with Walsh coefficients $S(w)\geqε$ with probability at least $1-δ$. However, in this paper, a quantum algorithm for this problem is given with query complexity $O(\frac{\log\frac{1}δ}{ε^4})$, which is independent of $n$. Furthermore, the quantum algorithm is generalized to apply for an $n$ variable $m$ output Boolean function $F$ with query complexity $O(2^m\frac{\log\frac{1}δ}{ε^4})$.