On the Power of Non-Adaptive Learning Graphs

TL;DR

This paper formalizes the quantum query complexity of certificate structures, introduces a dual formulation of non-adaptive learning graphs, and demonstrates their optimality for certain problems, including the triangle and triangle-sum problems.

Contribution

It develops a dual formulation of non-adaptive learning graphs and proves their tightness for all certificate structures, extending quantum query bounds to new problems.

Findings

Non-adaptive learning graphs are tight for all certificate structures.

Constructs functions with certificate structures using orthogonal arrays.

Provides near-optimal quantum query bounds for the triangle and triangle-sum problems.

Abstract

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the $k$-sum problem derived recently in arXiv:1206.6528. Finally, we use these results to show that the learning graph for the triangle problem from arXiv:1210.1014 is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.