The classical discrete fourier transform takes a sequence of values and outputs another sequence of values that describe a set of coefficients for complex sinusoids which can be used to reconstruct (or approximate) the original input.
In contrast, what exactly does a quantum fourier transform do? And how is it said to be the "analogue" of the discrete fourier transform?
More specifically, since all the input qubits are in superposition with each other, is there still the notion of an ordered sequence as there is in DFT (since the order of the inputs of DFT clearly affects the output)?
And how does one interpret the output of QFT? Say the output of a QFT is a superposition of $a |00\rangle +b|01\rangle + c|10\rangle + d|11\rangle$, what does this tell me about the original superposition?