Quantum property testing for bounded-degree graphs

TL;DR

This paper introduces quantum algorithms that significantly improve the efficiency of testing bipartiteness and expansion in bounded-degree graphs, surpassing classical bounds, and establishes quantum lower bounds indicating limits of speedup.

Contribution

It presents the first quantum algorithms for these graph properties with sublinear time complexity and proves lower bounds, clarifying the potential and limits of quantum speedups in property testing.

Findings

Quantum algorithms solve bipartiteness and expansion in O(N^(1/3)) time.

Quantum lower bounds show no exponential speedup is possible for expansion testing.

Classical lower bound for bipartiteness testing is Omega(sqrt(N)).

Abstract

We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion, we also prove an Omega(N^(1/4)) quantum query lower bound, thus ruling out the possibility of an exponential quantum speedup. Our quantum algorithms follow from a combination of classical property testing techniques due to Goldreich and Ron, derandomization, and the quantum algorithm for element distinctness. The quantum lower bound is obtained by the polynomial method, using novel algebraic techniques and combinatorial analysis to accommodate the graph structure.