Evaluation of boolean formulas with restricted inputs

Abstract

In this thesis, I will investigate the running time of quantum algorithms for evaluating boolean functions when the input is promised to satisfy certain conditions. The two quantum algorithms considered in this paper are the quantum walk algorithm for NAND trees given by Farhi and Gutmann [2], and an algorithm for more general boolean formulas based on span programs, given by Reichardt and Spalek [6]. I will show that these algorithms can run much faster on a certain set of inputs, and that there is a super-polynomial separation between the quantum algorithm and the classical lower bound on this problem. I will apply this analysis to quantum walks on decision trees, as described in [3], giving a class of decision trees that can be penetrated quickly by quantum walk but may not be efficiently searchable by classical algorithms.