Quantum Sketches, Hashing, and Approximate Nearest Neighbors
Sajjad Hashemian

TL;DR
This paper proves that quantum sketches cannot compress large datasets for approximate nearest neighbor search below linear qubit size, but quantum algorithms can still offer quadratic speedups in query time.
Contribution
It establishes fundamental lower bounds on quantum memory for ANN data structures, showing limitations of quantum sketches in high-dimensional spaces.
Findings
Quantum sketches require linear qubits for certain datasets.
Amplitude amplification provides near-optimal quadratic speedups.
Lower bounds are proven via reductions to quantum random access codes.
Abstract
Motivated by Johnson--Lindenstrauss dimension reduction, amplitude encoding, and the view of measurements as hash-like primitives, one might hope to compress an -point approximate nearest neighbor (ANN) data structure into qubits. We rule out this possibility in a broad quantum sketch model, the dataset is encoded as an -qubit state , and each query is answered by an arbitrary query-dependent measurement on a fresh copy of . For every approximation factor and constant success probability , we exhibit -point instances in Hamming space with for which any such sketch requires 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,…
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Stochastic Gradient Optimization Techniques · Complexity and Algorithms in Graphs
