Quantum and Randomised Algorithms for Non-linearity Estimation

TL;DR

This paper introduces efficient quantum and randomized algorithms to approximate the non-linearity of Boolean functions with low query complexity, significantly improving over naive methods and establishing near-optimal bounds.

Contribution

It presents novel quantum and randomized algorithms for estimating Boolean function non-linearity with polynomial and constant query complexities, respectively, advancing the state of the art.

Findings

Randomized algorithm has linear query complexity in n.

Quantum algorithm's query complexity is independent of n.

Lower bounds show the algorithms are near-optimal.

Abstract

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires $\Theta(2^n)$ queries to an $n$-bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted $\lambda$, with query complexities that depend polynomially on $\lambda$. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in $n$, already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of $n$. Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique.