Quantum algorithms for testing Boolean functions

TL;DR

This paper explores quantum algorithms, especially Bernstein-Vazirani and Grover search, for efficiently identifying variable dependencies in Boolean functions, improving understanding of quantum query complexity.

Contribution

It demonstrates how quantum algorithms can deterministically identify linear Boolean functions and probabilistically learn variable dependencies in general Boolean functions.

Findings

Deterministically identifies linear Boolean functions with one query

Learns variable dependencies with success probability independent of input size

Proposes amplification of success probability using Grover search

Abstract

We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables; given a linear Boolean function, the Bernstein-Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. The same quantum algorithm can also be used to learn which input variables other types of Boolean functions depend on, with a success probability that depends on the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure to futher amplify the success probability, based on another quantum algorithm, the Grover search.