Solution
= Solution
In the final state, the second-register amplitude vector associated with outcome $y$ is
$$
2^{-n}\sum_x(-1)^{x\cdot y}|f(x)\rangle.
$$
For injective $f$, part (ii) says that its squared norm is $2^{-2n}2^n=2^{-n}$. By the <Born rule for a product-basis measurement>, every $y\in B_n$ is therefore observed with probability $2^{-n}$: the input register is uniformly distributed.