# Quantum Algorithms for Classical Probability Distributions

## TL;DR

This paper explores quantum algorithms for classical probability distributions, establishing models for quantum access, and demonstrating a quadratic speedup in distinguishing distributions via the inverse Hellinger distance.

## Contribution

It introduces four models for quantum access to classical distributions and proves a quadratic quantum advantage in distribution distinguishability using the adversary method.

## Key findings

- Quantum query complexity is inversely proportional to Hellinger distance.
- Four models for quantum access to classical distributions are formulated.
- Quantum algorithms can distinguish distributions more efficiently than classical methods.

## Abstract

We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships. Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions. The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.