Quantum tests for the linearity and permutation invariance of Boolean functions

TL;DR

This paper introduces two quantum algorithms for property testing of Boolean functions, efficiently determining linearity and permutation invariance with fewer oracle calls than classical methods.

Contribution

The paper presents novel quantum algorithms that improve the efficiency of testing Boolean function properties, specifically linearity and symmetry, using fewer oracle calls than classical algorithms.

Findings

Quantum algorithms require O(epsilon^{-2/3}) calls, outperforming classical methods.

The linearity test can identify the specific linear function if the function is linear.

Uses Bernstein-Vazirani algorithm, amplitude amplification, and projective measurements.

Abstract

The goal in function property testing is to determine whether a black-box Boolean function has a certain property or is epsilon-far from having that property. The performance of the algorithm is judged by how many calls need to be made to the black box in order to determine, with high probability, which of the two alternatives is the case. Here we present two quantum algorithms, the first to determine whether the function is linear and the second to determine whether it is symmetric (invariant under permutations of the arguments). Both require O(epsilon^{-2/3}) calls to the oracle, which is better than known classical algorithms. In addition, in the case of linearity testing, if the function is linear, the quantum algorithm identifies which linear function it is. The linearity test combines the Bernstein-Vazirani algorithm and amplitude amplification, while the test to determine whether a function is symmetric uses projective measurements and amplitude amplification.