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6

In the paper I called it the Burnside decomposition, but it looks like the standard name is the Wedderburn decomposition. That might simply have been a mistake in terminology on my part. Anyway, there are two good ways to get the summands to be $V \otimes V^*$. (Of course they are closely related.) 1) You can interpret $\mathbb{C}[G]$ as an associative ...


4

To see that Simon's program is an instance of an (abelian) hidden subgroup problem, we have to identify the group $G$, the subgroup $H$, the set $X$ and the function $f : G \rightarrow X$. Note first that the set $\{ 0,1 \}^n$ of all bit vectors of length $n$ naturally comes with a group structure given by the (component-wise) XOR between bit vectors: $(x_1, ...


3

The second part of each tensor product serves as a multiplicity space. It might be more satisfying to write it as a full decomposition like your second one. So you have $\bigoplus V_\lambda \otimes V_\lambda^*$ and you want it to look more like your second equation. What would happen is each $V_\lambda$ would show up as a direct summand $d_\lambda$ times ...


3

The question is whether taking the Fourier transform $\operatorname{QFT}|gH\rangle$ followed by sampling allows to efficiently recover generators of the hidden subgroup $H\leq G$. While the problem is wide open for non-abelian groups (see this paper for a discussion of the limitations of the Fourier sampling method for instances in case of $G=S_n$, $G=PSL(2,\...


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