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Most quantum algorithms that give an exponential speedup over classicalalgorithms exploit the Fourier transform in some way. In Shor's algorithm,sampling from the quantum Fourier spectrum is used to discover periodicity ofthe modular exponentiation function. In a generalization of this idea, quantumFourier sampling can be used to discover hidden subgroup structures of somefunctions much more efficiently than it is possible classically. Anotherproblem for which the Fourier transform has been recruited successfully on aquantum computer is the hidden shift problem. Quantum algorithms for hiddenshift problems usually have a slightly different flavor from hidden subgroupalgorithms, as they use the Fourier transform to perform a correlation with agiven reference function, instead of sampling from the Fourier spectrumdirectly. In this paper we show that hidden shifts can be extracted efficientlyfrom Boolean functions that are quadratic forms. We also show how to identifyan unknown quadratic form on n variables using a linear number of queries, incontrast to the classical case were this takes Theta(n^2) many queries to ablack box. What is more, we show that our quantum algorithm is robust in thesense that it can also infer the shift if the function is close to a quadratic,where we consider a Boolean function to be close to a quadratic if it has alarge Gowers U_3 norm.

Quantum algorithms to solve the hidden shift problem for quadratics and for functions of large Gowers norm