One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e. g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient.
Soon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi <i>et. al. </i> [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can.
I begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework.
UPDATE (February 2003): Please see article http://arxiv. org/abs/quant-ph/0302138 for a resolution to the problem of simulating the continuous-time adiabatic search algorithm with a quantum circuit using only O(sqrt(N)) resources.
Identifer | oai:union.ndltd.org:WATERLOO/oai:uwspace.uwaterloo.ca:10012/1129 |
Date | January 2002 |
Creators | Ioannou, Lawrence |
Publisher | University of Waterloo |
Source Sets | University of Waterloo Electronic Theses Repository |
Language | English |
Detected Language | English |
Type | Thesis or Dissertation |
Format | application/pdf, 681507 bytes, application/pdf |
Rights | Copyright: 2002, Ioannou, Lawrence. All rights reserved. |
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