Quantum algorithms for learning a hidden graph and beyond

TL;DR

This paper develops quantum algorithms for learning unknown graphs more efficiently than classical methods in certain query models, achieving exponential speedups in some cases and establishing limits of quantum advantage.

Contribution

It introduces quantum algorithms for graph learning in various query models, demonstrating significant speedups and providing new techniques for combinatorial group testing and function learning.

Findings

Quantum algorithms outperform classical in OR and parity models for some graphs.

Exponential speedups are achievable in the parity query model under certain conditions.

No quantum speedup is possible in the OR model without graph promises.

Abstract

We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any edges. In the second ("parity queries"), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al.\ for a "gapped" version of the group testing problem. We also give simple time-efficient quantum algorithms based on Fourier sampling and amplitude amplification for learning the exact-half and majority functions, which almost match the optimal complexity of Belovs' algorithms.