A Fast Quantum Algorithm for the Affine Boolean Function Identification

TL;DR

This paper presents a quantum algorithm that efficiently identifies affine Boolean functions with a minimal number of queries, extending the Bernstein-Vazirani algorithm to affine and incompletely defined functions with high probability.

Contribution

It introduces a two-query quantum algorithm for affine Boolean functions and demonstrates bounded-error algorithms for incompletely defined functions, improving identification efficiency.

Findings

Two-query algorithm identifies affine functions with certainty

Bounded-error algorithms achieve at least 2/3 success probability

Extension of Bernstein-Vazirani algorithm to affine and incomplete functions

Abstract

Bernstein-Vazirani algorithm (the one-query algorithm) can identify a completely specified linear Boolean function using a single query to the oracle with certainty. The first aim of the paper is to show that if the provided Boolean function is affine, then one more query to the oracle (the two-query algorithm) is required to identify the affinity of the function with certainty. The second aim of the paper is to show that if the provided Boolean function is incompletely defined, then the one-query and the two-query algorithms can be used as bounded-error quantum polynomial algorithms to identify certain classes of incompletely defined linear and affine Boolean functions respectively with probability of success at least $2/3$.