量子草图、哈希与近似最近邻
Quantum Sketches, Hashing, and Approximate Nearest Neighbors
AI总结：
该研究排除了将近似最近邻数据结构压缩到O(log n)量子比特的可能性，并展示了量子方法在候选扫描中的优势。
摘要：
受Johnson-Lindenstrauss维度降低、振幅编码以及将测量视为哈希样原语的观点启发，人们可能希望将一个n点的近似最近邻(ANN)数据结构压缩成O(log n)个量子比特。我们在一个广泛的量子草图模型中排除了这种可能性，数据集P被编码为一个m个量子比特的状态ρ_P，每个查询通过在ρ_P的一个新鲜副本上的任意查询依赖测量来回答。对于每个近似因子c≥1和常数成功概率p>1/2，我们展示了一个在汉明空间{0,1}^d中的n点实例，其中d=Θ(log n)，任何这样的草图都需要m=Ω(n)个量子比特，通过将其减少到量子随机访问码和Nayak的下界。这些内存下界与潜在的量子查询时间增益和基于哈希的ANN的候选扫描抽象共存，在候选扫描抽象中，振幅放大导致候选检查的平方减少，这通过Grover/BBBV型界限本质上是最优的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
