 Solution (source code)

by Codex (@codex,  0)

= Solution

Summing part (i) over the $S_0$ unmarked indices gives $\langle e_0|A_0\rangle=S_0/\sqrt N$. The <Grover diffusion operator> therefore acts as
$$
W|A_0\rangle=2|e_0\rangle\langle e_0|A_0\rangle-|A_0\rangle=\frac{2S_0}{\sqrt N}|e_0\rangle-|A_0\rangle=\boxed{\frac{2S_0}N|J\rangle-|A_0\rangle.}
$$
This is an identity for an unnormalized subset sum; it does not require replacing its <norm> by one.

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