Quantum search in a dictionary based on fingerprinting-hashing

TL;DR

This paper devises a quantum dictionary search that leverages quantum fingerprinting-hashing to compress the hash state and provide an initial amplitude amplification before Grover steps. The core Algorithm ${\cal A}$ achieves $O(\sqrt{n})$ oracle queries with memory $O(\log n + \log m)$ by encoding the vocabulary with a fingerprinting hash $\psi_E$ and applying a two-stage amplification and measurement. It further generalizes to Algorithm ${\cal A}2$ using arbitrary $(m,\epsilon,s)$-quantum hash functions, maintaining $O(\sqrt{n})$ queries while reducing memory to $\log n + s$, with good hash constructions (e.g., fingerprinting-based or Freivald fingerprinting) yielding $s = O(\log m)$. The results demonstrate memory-efficient quantum search in unstructured databases and lay groundwork for practical implementations of hashing-based quantum information retrieval. Overall, the work shows that quantum hashing can dramatically lower qubit resources required for large-scale search tasks while preserving quadratic speedups.

Abstract

In this work, we present a quantum query algorithm for searching a word of length $m$ in an unsorted dictionary of size $n$. The algorithm uses $O(\sqrt{n})$ queries (Grover operators), like previously known algorithms. What is new is that the algorithm is based on the quantum fingerprinting-hashing technique, which (a) provides a first level of amplitude amplification before applying the sequence of Grover amplitude amplification operators and (b) makes the algorithm more efficient in terms of memory use -- it requires $O(\log n + \log m)$ qubits. Note that previously developed algorithms by other researchers without hashing require $O(\log n + m)$ qubits.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Lemma 1
• Definition 1
• Definition 2
• Theorem 3
• Theorem 4