Quantum Sketches, Hashing, and Approximate Nearest Neighbors

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 $n$-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 $\rho_P$, and each query is answered by an arbitrary query-dependent measurement on a fresh copy of $\rho_P$. 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 $d=\Theta(\log n)$ for which any such sketch requires $m=\Omega(n)$ 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.