Leveraging Unknown Structure in Quantum Query Algorithms

TL;DR

This paper develops quantum query algorithms that maintain their efficiency even without prior knowledge of input structure, extending span program techniques to more general problems like state conversion.

Contribution

It introduces a modified span program algorithm that achieves structure-based speed-ups without pre-promises, applicable to problems like graph connectivity and state conversion.

Findings

Achieves $O( ilde{\sqrt{k}} n)$ queries without knowing $k$ in advance.

Extends span program algorithms to unpromised input structures.

Demonstrates persistent speed-ups in quantum algorithms without input promises.

Abstract

Quantum span program algorithms for function evaluation commonly have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these speed-ups persist even without having a promise ahead of time, and we extend this approach to the more general problem of state conversion. For example, there is a span program algorithm that decides whether two vertices are connected in an $n$-vertex graph with $O(n^{3/2})$ queries in general, but with $O(\sqrt{k}n)$ queries if promised that, if there is a path, there is one with at most $k$ edges. Our algorithm uses $\tilde{O}(\sqrt{k}n)$ queries to solve this problem if there is a path with at most $k$ edges, without knowing $k$ ahead of time.