# Hidden Subgroup Problem: Embedding $G$ in a complex hilbert space $H$

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.

• If there is a nexus of your Question with the subject of cryptography, it would improve your post to include a sketch of that context. Perhaps you have in mind (for finite abelian groups) the use of Schor's algorithm for factoring with quantum computers? Cf. here. Jun 8 at 16:05
• Or, is it about the details of how Schor's algorithm works (when you view it as an operation within the complex Hilbert space)? Jun 8 at 16:30
• @hardmath I'm studying the hidden subgroup problem in general, but i guess a particular example that would be enlightening would be how we can apply QFT to efficiently solve the discrete log problem? Jun 9 at 1:27
• Note also that sister site Theoretical Computer Science has a fair number of posts concerning the hidden subgroup problem and its applications to complexity. Jun 9 at 13:39

$$\rho: G\to \mathsf{End}(V)$$ that behave well with respect to the group operations, i.e. $$\rho(\mathsf{id}_G) = \mathsf{id}_V, \rho(gg') = \rho(g)\rho(g'), \rho(g^{-1}) = \rho(g)^{-1}.$$
Here, $$\mathsf{End}(V)$$ is the group of endomorphisms of $$V$$, or equivalently for $$\dim V = n$$ it is $$k^{n\times n}$$, viewed as a group of matrices.