Almost all decision trees do not allow significant quantum speed-up

TL;DR

Most decision trees of limited depth require linear quantum queries, indicating that classical algorithms in this model do not benefit significantly from quantum speed-up, due to high average sensitivity.

Contribution

This work proves that nearly all decision trees of bounded depth have high quantum query complexity, limiting quantum advantages in this class.

Findings

Most decision trees require Omega(d) quantum queries

High average sensitivity of random decision trees

Classical algorithms in query model lack significant quantum speed-up

Abstract

We show that, for any d, all but a doubly exponentially small fraction of decision trees of depth at most d require Omega(d) quantum queries to be computed with bounded error. In other words, most efficient classical algorithms in the query complexity model do not admit a significant quantum speed-up. The proof is based on showing that, with high probability, the average sensitivity of a random decision tree is high.