  Most quantum algorithms that give an exponential speedup over classical
algorithms exploit the Fourier transform in some way. In Shor's algorithm,
sampling from the quantum Fourier spectrum is used to discover periodicity of
the modular exponentiation function. In a generalization of this idea, quantum
Fourier sampling can be used to discover hidden subgroup structures of some
functions much more efficiently than it is possible classically. Another
problem for which the Fourier transform has been recruited successfully on a
quantum computer is the hidden shift problem. Quantum algorithms for hidden
shift problems usually have a slightly different flavor from hidden subgroup
algorithms, as they use the Fourier transform to perform a correlation with a
given reference function, instead of sampling from the Fourier spectrum
directly. In this paper we show that hidden shifts can be extracted efficiently
from Boolean functions that are quadratic forms. We also show how to identify
an unknown quadratic form on n variables using a linear number of queries, in
contrast to the classical case were this takes Theta(n^2) many queries to a
black box. What is more, we show that our quantum algorithm is robust in the
sense 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 a
large Gowers U_3 norm.
