The Exercise

The exercise belongs to the hidden subgroup problem on a finite abelian group $G = \mathbb{Z}_{n_1} \times \cdots \times \mathbb{Z}_{n_M}$, a product of cyclic groups. The quantum Fourier transform over $G$, applied to a uniform superposition over a coset of the hidden subgroup $K$, gives each outcome $\ell \in G$ an amplitude proportional to a sum, over the elements of $K$, of the character of $G$ labelled by $\ell$. The exercise takes a hidden subgroup that splits along the same cyclic factors as $G$, and asks to reduce that sum, and the condition that it equal the order $|K|$ of $K$, to the separate factors.

Characters and Product Subgroups

An element $g$ of $G$ is a tuple $(g_1, \dots, g_M)$ with each $g_j$ in $\mathbb{Z}_{n_j}$. The character $\chi_\ell$ of $G$ labelled by $\ell \in G$ is the complex function on $G$ $$\chi_\ell(g) = \prod_{j=1}^{M} e^{2\pi i \, \ell_j g_j / n_j},$$(1) the product over the factors of the standard character $x \mapsto e^{2\pi i x / n_j}$ of $\mathbb{Z}_{n_j}$, evaluated at the product $\ell_j g_j$ taken in $\mathbb{Z}_{n_j}$.

Let $K_j$ be a subgroup of $\mathbb{Z}_{n_j}$ for each $j$. Their product subgroup $K = K_1 \times \cdots \times K_M$ is the subgroup of $G$ of the elements $h$ with $h_j \in K_j$ for every $j$.

The Claim

What Is Checked

The equivalence (3) is stated over the product subgroup $K$ of the subgroups $K_j$, an element $\ell$ of $G$, and the standard characters of the cyclic factors, with $|K|$ and $|K_j|$ the orders of the groups. What the library proves about the product subgroup is the factorization (2) of a character sum over it.