Motivated by Johnson--Lindenstrauss dimension reduction, amplitude encoding, and the view of measurements as hash-like primitives, one might hope to compress an $n$-point approximate nearest neighbor (ANN) data structure into $O(\log n)$ qubits. We rule out this possibility in a broad quantum sketch model, the dataset $P$ is encoded as an $m$-qubit state $ρ_P$, and each query is answered by an arbitrary query-dependent measurement on a fresh copy of $ρ_P$. For every approximation factor $c\ge 1$ and constant success probability $p>1/2$, we exhibit $n$-point instances in Hamming space $\{0,1\}^d$ with $d=Θ(\log n)$ for which any such sketch requires $m=Ω(n)$ qubits, via a reduction to quantum random access codes and Nayak's lower bound. These memory lower bounds coexist with potential quantum query-time gains and in candidate-scanning abstractions of hashing-based ANN, amplitude amplification yields a quadratic reduction in candidate checks, which is essentially optimal by Grover/BBBV-type bounds.

翻译：受Johnson-Lindenstrauss降维、振幅编码以及将测量视为类哈希原语的启发，人们可能期望将$n$点近似最近邻（ANN）数据结构压缩至$O(\log n)$量子比特。我们在广义量子草图模型中排除了这种可能性：数据集$P$被编码为$m$量子比特态$ρ_P$，每个查询通过对$ρ_P$的新副本进行任意查询相关测量来回答。对于每个近似因子$c\ge 1$和恒定成功概率$p>1/2$，我们通过归约至量子随机访问码和Nayak下界，在汉明空间$\{0,1\}^d$（其中$d=Θ(\log n)$）中构造了$n$点实例，证明任何此类草图需要$m=Ω(n)$量子比特。这些内存下界与潜在的量子查询时间增益共存：在基于哈希的ANN的候选扫描抽象模型中，振幅放大可实现候选检查次数的二次缩减，这通过Grover/BBBV类下界证明本质是最优的。