Quantum Approximate Counting for Markov Chains and Application to Collision Counting

TL;DR

This paper extends quantum approximate counting techniques to Markov chains, enabling more efficient collision counting algorithms with improved query complexity over classical methods in certain cases.

Contribution

It generalizes quantum approximate counting to Markov chains and applies this to improve collision counting algorithms using quantum walks.

Findings

Achieves quantum collision counting with fewer queries than classical methods.

Provides a quantum algorithm estimating the number of collisions with relative error.

Demonstrates improved efficiency for small collision counts (m) relative to N.

Abstract

In this paper we show how to generalize the quantum approximate counting technique developed by Brassard, H{\o}yer and Tapp [ICALP 1998] to a more general setting: estimating the number of marked states of a Markov chain (a Markov chain can be seen as a random walk over a graph with weighted edges). This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful "quantum walk based search" framework established by Magniez, Nayak, Roland and Santha [SIAM Journal on Computing 2011]. As an application, we apply this approach to the quantum element distinctness algorithm by Ambainis [SIAM Journal on Computing 2007]: for two injective functions over a set of $N$ elements, we obtain a quantum algorithm that estimates the number $m$ of collisions of the two functions within relative error $\epsilon$ by making $\tilde{O}\left(\frac{1}{\epsilon^{25/24}}\big(\frac{N}{\sqrt{m}}\big)^{2/3}\right)$ queries, which gives an improvement over the $\Theta\big(\frac{1}{\epsilon}\frac{N}{\sqrt{m}}\big)$-query classical algorithm based on random sampling when $m\ll N$.