A question about performing quantum computations on uniform superpositions

Question

Let us consider the following situation. Let $U_f$ be a gate computing $f$ mapping $\{0,1\}^n$ to $\{0,1\}^n$. That is, $U_f\left\vert x,0^n\right\rangle=\left\vert x,f(x)\right\rangle$. Let $\left\vert\phi\right\rangle$ be the uniform superposition on $\{0,1\}^n$. By performing $U_f$ on $\left\vert\phi\right\rangle\left\vert0^n\right\rangle$, we have $\left\vert\phi'\right\rangle=\sum_{x\in\{0,1\}^n}\frac1{2^{n/2}}\left\vert x,f(x)\right\rangle$. Let $x^\ast$ be some specific state $x^\ast\in\{0,1\}^n$.

My question is: is it possible to obtain $f(x^\ast)$ from performing some gates or projections on $\left\vert\phi'\right\rangle$ (without running $U_f$ again) with overwhelming probability? Or, particularly, is it possible to obtain $f(0^n)$ from $\left\vert\phi'\right\rangle$? Does Hadamard gate work in this situation?

I guess no, but I wonder there is something I have missed.

Answer 1

You could run Grover's algorithm on the top $n$ bits of the register for $2^{n/2}$ steps, but this is probably less efficient than you were hoping for.

Anything better than Grover is unlikely to work (I'm not sure how far Zalka's no-go result in "Grover's quantum searching algorithm is optimal" extends into the following). Such an algorithm would be enough to invert an arbitrary permutation on $\mathbb F_2^n$ (and hence any permutation as a corollary). To see this suppose we are endowed with a circuit $U_\pi$ to evaluate our mystery permutation $\pi(x)$. We create the state $|\phi\rangle|0^n\rangle$ and apply $U_\pi$ to obtain $|\psi\rangle:=\sum 2^{-n/2}|x\rangle|\pi(x)\rangle$. Note that the final $n$ bits are in state $|\phi\rangle$ because $\pi$ is a permutation. If swap the first and second half of the registers we then have $\sum 2^{-n/2}|x\rangle|\pi^{-1}(x)\rangle$ and running our putative algorithm for your problem will allow us to compute $\pi^{-1}(x^*)$ for any $x^*$. Restricting to the case $0^n$ does to reduce the power of such an algorithm by considering the function (resp. permutation) $f(x\oplus x^*)$ (resp. $\pi(x)\oplus x^*$).