In general or in specific examples, how is the group used in an instance of the hidden subgroup problem embedded into a complex Hilbert space in order to apply the quantum fourier transform needed to solve the problem efficiently?
In quantum computing we imagine qubits as unit vectors in a complex Hilbert space $\mathbb{C}^{2^n}$ where $n$ is the number of bits in the system.
I've thought about this myself and come up with the idea that we can decompose a group into a product of cyclic groups, which can then be embedded as "rings" around the unit n-sphere in $\mathbb{C}^{2^n}$, where each cyclic group gets its own ring about a plane, but im not sure if this is the "usual" way this is done.