On the uselessness of quantum queries

TL;DR

This paper establishes a fundamental connection between classical and quantum query uselessness, showing that classical useless queries imply quantum uselessness, and provides a new method for lower bounds on quantum query complexity.

Contribution

It proves that if classical queries are useless for a problem, then a proportional number of quantum queries are also useless, extending the concept beyond Boolean functions.

Findings

Classical useless queries imply quantum uselessness for the same problem.

The result applies to non-Boolean functions and properties.

Provides a new approach to lower bounds in quantum query complexity.

Abstract

Given a prior probability distribution over a set of possible oracle functions, we define a number of queries to be useless for determining some property of the function if the probability that the function has the property is unchanged after the oracle responds to the queries. A familiar example is the parity of a uniformly random Boolean-valued function over $\{1,2,...,N\}$, for which $N-1$ classical queries are useless. We prove that if $2k$ classical queries are useless for some oracle problem, then $k$ quantum queries are also useless. For such problems, which include classical threshold secret sharing schemes, our result also gives a new way to obtain a lower bound on the quantum query complexity, even in cases where neither the function nor the property to be determined is Boolean.