The quantum complexity of approximating the frequency moments

TL;DR

This paper investigates the quantum computational complexity of approximating frequency moments of data sequences, demonstrating quantum algorithms that outperform classical methods in query and streaming models.

Contribution

It introduces new quantum algorithms for frequency moments that surpass classical algorithms in query complexity and streaming space efficiency.

Findings

Quantum algorithms outperform classical ones in query complexity for frequency moments.

Quantum streaming algorithms use less space than classical counterparts for certain parameters.

Quantum methods achieve significant improvements in approximating specific frequency moments.

Abstract

The $k$'th frequency moment of a sequence of integers is defined as $F_k = \sum_j n_j^k$, where $n_j$ is the number of times that $j$ occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate $F_k$ up to relative error $\epsilon$. We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements of the input one at a time, and seek to minimise the amount of storage space, or passes over the data, used to approximate $F_k$. We describe quantum algorithms for $F_0$, $F_2$ and $F_\infty$ in this model which substantially outperform the best possible classical algorithms in certain parameter regimes.