Quantum lower bound for inverting a permutation with advice

Given a random permutation $f: [N] \to [N]$ as a black box and $y \in [N]$, we want to output $x = f^{-1}(y)$. Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but \emph{not} on the input $y$. Classically, there is a data structure of size $\tilde{O}(S)$ and an algorithm that with the help of the data structure, given $f(x)$, can invert $f$ in time $\tilde{O}(T)$, for every choice of parameters $S$, $T$, such that $S\cdot T \ge N$. We prove a quantum lower bound of $T^2\cdot S \ge \tildeΩ(εN)$ for quantum algorithms that invert a random permutation $f$ on an $ε$ fraction of inputs, where $T$ is the number of queries to $f$ and $S$ is the amount of advice. This answers an open question of De et al. We also give a $Ω(\sqrt{N/m})$ quantum lower bound for the simpler but related Yao's box problem, which is the problem of recovering a bit $x_j$, given the ability to query an $N$-bit string $x$ at any index except the $j$-th, and also given $m$ bits of advice that depend on $x$ but not on $j$.

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