Quantum tests for the linearity and permutation invariance of Boolean functions

Understand

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.