Efficient Quantum Algorithms related to Autocorrelation Spectrum

TL;DR

This paper introduces efficient quantum algorithms for sampling and estimating autocorrelation spectrum properties of Boolean functions, extending the Deutsch-Jozsa algorithm to higher-order derivatives and providing practical estimation methods.

Contribution

It develops novel quantum algorithms for autocorrelation spectrum analysis, including sampling from Walsh spectra of derivatives and estimating autocorrelation coefficients, advancing quantum analysis tools.

Findings

Quantum algorithms for Walsh spectrum sampling of derivatives

Sampling input points based on autocorrelation coefficients

Estimating autocorrelation coefficient squares efficiently

Abstract

In this paper, we propose efficient probabilistic algorithms for several problems regarding the autocorrelation spectrum. First, we present a quantum algorithm that samples from the Walsh spectrum of any derivative of $f()$. Informally, the autocorrelation coefficient of a Boolean function $f()$ at some point $a$ measures the average correlation among the values $f(x)$ and $f(x \oplus a)$. The derivative of a Boolean function is an extension of autocorrelation to correlation among multiple values of $f()$. The Walsh spectrum is well-studied primarily due to its connection to the quantum circuit for the Deutsch-Jozsa problem. We extend the idea to "Higher-order Deutsch-Jozsa" quantum algorithm to obtain points corresponding to large absolute values in the Walsh spectrum of a certain derivative of $f()$. Further, we design an algorithm to sample the input points according to squares of the autocorrelation coefficients. Finally we provide a different set of algorithms for estimating the square of a particular coefficient or cumulative sum of their squares.