Nonadaptive quantum query complexity

TL;DR

This paper investigates the limitations of nonadaptive quantum query algorithms, demonstrating they require linear queries for total functions and only offer limited speed-up over classical algorithms in learning problems.

Contribution

It establishes lower bounds on nonadaptive quantum query complexity and compares quantum and classical query efficiencies for learning problems.

Findings

Nonadaptive quantum algorithms need Omega(n) queries for total functions.

Quantum algorithms offer limited speed-up over classical algorithms in nonadaptive learning.

Classical algorithms can match quantum performance with O(k log m) queries.

Abstract

We study the power of nonadaptive quantum query algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum query algorithm that computes some total boolean function depending on n variables must make Omega(n) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using O(k log m) queries to solve the same problem. Thus, in the nonadaptive setting, quantum algorithms can achieve at most a very limited speed-up over classical query algorithms.