A polynomial quantum query lower bound for the set equality problem

TL;DR

This paper establishes a polynomial lower bound on the number of quantum queries needed to solve the set equality problem, which is related to graph isomorphism, advancing understanding of quantum query complexity.

Contribution

It provides the first non-trivial polynomial lower bound for quantum algorithms solving the set equality problem under the promise.

Findings

Quantum query complexity lower bound is ( (n/\u2212ln n))^{1/5}

Sets of size n require at least (n/ ln n)^{1/5} quantum queries

Advances understanding of quantum limits for set equality and related problems

Abstract

The set equality problem is to tell whether two sets $A$ and $B$ are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any $\omega(1)$ query lower bound when sets $A$ and $B$ are given by quantum oracles. We will show that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles $\Omega(\sqrt[5]{\frac{n}{\ln n}})$ times, where $n=|A|=|B|$.