# The hidden subgroup problem for finite Abelian groups when $|G| \neq 2^n$

I’ve been studying Simon’s problem and developing simulation models using Mathematica to extend the problem to other abelian groups and hidden subgroups of order $$\geq 2$$. I can now deduce any “hidden” proper subgroup of AbelianGroup[{2,2,2}], AbelianGroup[{2,4}], or AbelianGroup[{8}].

I’m now trying to apply my simulation to Abelian groups of order 6, e.g. AbelianGroup[{2,3}] and AbelianGroup[{6}].

Can anyone cite where worked examples of HSP have been applied to groups with order not equal to $$2^n$$?

• Welcome to QCSE. The order of $G$ in Shor’s algorithm is $N$, the odd number to be factored, isn’t it? We do embed it and work in a larger ambient Hilbert space of size $2^n$, though. Commented Nov 15, 2023 at 12:03
• @MarkSpinelli yes, I know the full Hilbert space of n qubits is not used, and I’ve used variations on the Fourier transform, but nothing has worked properly as yet. Commented Nov 15, 2023 at 14:00
• @MarkSpinelli that is a subspace of n qubits normalized uniformly with 1/Sqrt(|G|). Commented Nov 15, 2023 at 14:11
• The Fourier transform is over $\mathbb Z/2^n \mathbb Z$ (even though the order of the group is less than $2^n$). That is, for Shor's algorithm the Fourier transform is the standard Fourier transform. Also if you think your question is more about Mathematica, then you might want to ask on mathematica.stackexchange.com (although there are some people on this site that are very familiar with Mathematica. Commented Nov 15, 2023 at 15:56
• @MarkSpinelli So when $|G| < 2^n$ the "full" Fourier transform is applied? After measurement are any of the (what would be $H^\perp$) thrown out as out of bounds to $G$? Commented Nov 15, 2023 at 23:32