Fourier 1-norm and quantum speed-up

TL;DR

This paper explores the relationship between Fourier 1-norm and quantum speed-up, providing bounds on classical vs. quantum query complexity and highlighting the role of Fourier analysis in quantum algorithms.

Contribution

It introduces a classical simulation method for quantum output probabilities based on Fourier 1-norm, linking it to quantum parallelism and query complexity bounds.

Findings

Fourier 1-norm influences quantum speed-up bounds

Classical simulation error depends on Fourier 1-norm

Bounds relate quantum and classical query complexities

Abstract

Understanding quantum speed-up over classical computing is fundamental for the development of efficient quantum algorithms. In this paper, we study such problem within the framework of the Quantum Query Model, which represents the probability of output $x \in \{0,1\}^n$ as a function $\pi(x)$. We present a classical simulation for output probabilities $\pi$, whose error depends on the Fourier $1$-norm of $\pi$. Such dependence implies upper-bounds for the quotient between the number of queries applied by an optimal classical algorithm and our quantum algorithm, respectively. These upper-bounds show a strong relation between Fourier $1$-norm and quantum parallelism. We show applications to query complexity.