Random Garbage Creates Exponential Quantum Query Separations

Abstract

We prove an exponential separation between standard (XOR) quantum queries and forward-erasing queries for a Boolean property of injective functions, answering a question of Aaronson~\cite{Aaronson2021Open}. We also obtain an exponential separation for a Boolean promise problem on permutations, where forward in-place queries are unitary but inverse queries are not supplied. Our construction follows Aaronson's suggestion of appending fixed random garbage to a Simon function. Let $X=\{0,1{\}}^n$ and $N=2^n$. We use the injection $f_{h,r}(x)=(h(x),x,r_x)$, where $h:X\to X$ is promised to be either a permutation or a Simon function, and $r:X\to X$ is an arbitrary tag table that is reused on every query. The resulting problem is solvable with at most $n+2$ standard queries, but requires $\Theta (\sqrt{N})$ forward-erasing queries, matching classical collision sampling. Embedding these instances into permutations, with the copy of $x$ replaced by a single bit that keeps the embedding injective, gives a promise problem on permutations of $L=4N^2$ elements that requires $O(\log L)$ standard queries versus $\Theta (L^{1/4})$ forward in-place queries. The lower bounds use the recording method. We replace each query by a nearby contraction that records cells of a purified tag table. After $T$ queries, tracing out the purified tag table yields a sum of positive-semidefinite contributions, each depending on $h$ at no more than $T$ addresses. A query-by-query hybrid bounds the trace-norm difference between the actual and auxiliary final operators by $O(T/\sqrt{N})$. Random permutations and random Simon functions have the same restriction distribution on such a set unless it contains a hidden Simon pair, an event of probability $O(T^2/N)$. Consequently, the actual final states in the two cases remain close unless $T=\Omega (\sqrt{N})$, giving the required lower bounds.