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Dissertation/Thesis Abstract

Some Applications of Quantum Walks to a General Class of Searches and the Computation of Boolean Functions
by Cottrell, Seth S., Ph.D., New York University, 2014, 155; 3665129
Abstract (Summary)

In previous papers about searches on star graphs several patterns have been made apparent; the speed up only occurs when graphs are ''tuned'' so that their time step operators have degenerate eigenvalues, and only certain initial states are effective. More than that, the searches are never faster than order square root of N time. In this thesis the problem is defined rigorously, the causes for all of these patterns are identified, sufficient and necessary conditions for quadratic-speed searches for any connected subgraph are demonstrated, the tolerance of these conditions is investigated, and it is shown that (unfortunately) we can do no better than order square root of N time. Along the way, a useful formalism is established that may be useful in future work involving highly symmetric graphs.

The tools and techniques so derived are then used to demonstrate that tree graphs can be used for the computation of Boolean functions. The philosophy of Farhi's work on the continuous-time NAND tree is applied to a discrete-time walk with any (AND, OR, NAND, or NOR) gate at each vertex. Tentative results show that the vast majority of possible Boolean functions on N bits can be calculated in order square root of N time.

Indexing (document details)
Advisor: Hillery, Mark, Cappell, Sylvain
Commitee: Sleator, Tycho
School: New York University
Department: Mathematics
School Location: United States -- New York
Source: DAI-B 76/04(E), Dissertation Abstracts International
Subjects: Applied Mathematics, Mathematics, Quantum physics
Keywords: Quantum algorithm, Quantum computation, Quantum search, Quantum walk
Publication Number: 3665129
ISBN: 978-1-321-37432-2
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