Quantum walk based search methods and algorithmic applications
Katalin Friedl · 2014
This thesis provides an introduction to the quantum walk based search method which is the quantum analogue of the classical Markov chain based search. We develop the basic theory of the Markov chain based search together with its quantisation due to Mario Szegedy[Sz2]. We deeply analyse their relationship, and sketch some of the possible algorithmic applications. The thesis also addresses the question of the implementation, which is largely hidden by the high level description. We show how to implement the quantum walk based search algorithm in the case of the element distinctness problem. We implement the necessary operations using only very basic data structures, mainly arrays. Also while we use a weak and basic computational model we could still significantly reduce the exponent of the polylog terms arising from counting non oracle calls. Another interesting achievement is that we could parallelise the Setup operation reducing its circuit depth form O(N ) to O(polylog(N)). Probably the most important tool we introduced for the implementation is the use of reversible sorting networks, but we introduce some other tricks as well. The optimal quantum query algorithm for the element distinctness problem was first described by Andris Ambainis [Amb3] and provided the basis of the generalised method. In this seminal paper Ambainis also addressed the question of implementation. However that implementation method does not fully cast into the general framework which was introduced later. The operations described by Ambainis use some involved data structures, such as skip lists and hash table. Also it uses a strong computational model for example, a qRAM query has cost only 1. In contrast with Ambainis’s single node implementation method ours uses bipartite style approach fitting the general framework of Szegedy. At first sight it may seem unnecessary since it results in a duplication of a large amount of data stored during the walk steps, but in fact it simplifies the implementation process. On the structure of the thesis: Section 1 covers some background on quantum walks and basic techniques related to the quantum walk based search method. Section 2-5 gives a comprehensive description of the general scheme introduced by Szegedy and derives the main theorems showing the quadratic speed-up compared to the classical case. These Sections roughly follow the structure of Szegedy’s original paper [Sz2], but the proofs are restructured, so that I could state the main theorem in a slightly stronger form with a much improved constant. Section 6 addresses the implementation issues already mentioned. Finally, Section 7 gives a brief overview of the further generalised scheme introduced by Frederic Magniez, Ashwin Nayak, Jeremie Roland and Miklos Santha. Quantum walk based search methods and algorithmic applications 2