Super-Polynomial Quantum Speed-ups for Boolean Evaluation Trees with Hidden Structure

TL;DR

This paper presents a quantum algorithm that achieves super-polynomial speed-ups over classical methods for evaluating certain structured boolean formulas, exploiting hidden structural properties of the input to reduce query complexity.

Contribution

It introduces a quantum algorithm for evaluating boolean formulas with hidden structure, achieving super-polynomial speed-up over classical lower bounds.

Findings

Quantum algorithm evaluates depth n trees with $O(n^{2+ ext{log}\omega})$ queries.

Classical lower bound established at $n^{ ext{Omega}( ext{log}\log n)}$ queries.

Super-polynomial quantum speed-up demonstrated for structured boolean evaluation.

Abstract

We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth $n$ tree using $O(n^{2+\log\omega})$ queries, where $\omega$ is independent of $n$ and depends only on the type of subformulas within the tree. We also prove a classical lower bound of $n^{\Omega(\log\log n)}$ queries, thus showing a (small) super-polynomial speed-up.