Set membership with two classical and quantum bit probes

TL;DR

This paper investigates efficient set membership data structures using classical and quantum bit probes, providing new upper bounds on memory requirements for both models.

Contribution

It introduces improved upper bounds for classical and quantum set membership schemes with two probes, advancing understanding of their memory efficiency.

Findings

Classical scheme bounds improved for t=2 probes.

Quantum scheme bounds also improved, surpassing previous results.

Results demonstrate advantages of quantum approaches in set membership problems.

Abstract

We consider the following problem: Given a set S of at most n elements from a universe of size m, represent it in memory as a bit string so that membership queries of the form "Is x in S?" can be answered by making at most t probes into the bit string. Let s(m,n,t) be the minimum number of bits needed by any such scheme. We obtain new upper bounds for s(m,n,t=2), which match or improve all the previously known bounds. We also consider the quantum version of this problem and obtain improved upper bounds.