# Improved Quantum Query Complexity on Easier Inputs

## TL;DR

This paper develops a modified quantum span program algorithm that maintains improved query complexity without input promises and demonstrates exponential quantum advantages in average query complexity for certain search problems.

## Contribution

It introduces a new span program algorithm that extends previous promise-based improvements to unpromised inputs and applies it to show significant quantum advantages in search tasks.

## Key findings

- Maintains query complexity improvements without input promises
- Achieves exponential quantum advantages in average query complexity
- Generalizes Montanaro's Search with Advice to broader problems

## Abstract

Quantum span program algorithms for function evaluation sometimes have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these improvements persist even without a promise ahead of time, and we extend this approach to the more general problem of state conversion. As an application, we prove exponential and superpolynomial quantum advantages in average query complexity for several search problems, generalizing Montanaro's Search with Advice [Montanaro, TQC 2010].