Quantum approximate counting for Markov chains and collision counting
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Published:2022-11
Issue:15&16
Volume:22
Page:1261
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ISSN:1533-7146
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Container-title:Quantum Information and Computation
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language:
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Short-container-title:QIC
Author:
Gall Francois Le,Ng Iu-Iong
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 $m\ll N$.
Subject
Computational Theory and Mathematics,General Physics and Astronomy,Mathematical Physics,Nuclear and High Energy Physics,Statistical and Nonlinear Physics,Theoretical Computer Science