Topics in quantum algorithms : adiabatic algorithm, quantum money, and bomb query complexity

Abstract

In this thesis, I present three results on quantum algorithms and their complexity. The first one is a numerical study on the quantum adiabatic algorithm( QAA) . We tested the performance of the QAA on random instances of MAX 2-SAT on 20 qubits and showed 3 strategics that improved QAA's performance, including a counter intuitive strategy of decreasing the overall evolution time. The second result is a security proof for the quantum money by knots proposed by Farhi et. al. We proved that quantum money by knots can not be cloned in a black box way unless graph isomorphism is efficiently solvable by a quantum computer. Lastly we defined a modified quantum query model, which we called bomb query complexity B(J), inspired by the Elitzur-Vaidman bomb-testing problem. We completely characterized bomb query complexity be showing that B(f) = [Theta](Q(f)2 ). This result implies a new method to find upper bounds on quantum query complexity, which we applied on the maximum bipartite matching problem to get an algorithm with O(n1.75) quantum query complexity, improving from the best known trivial O(n2 ) upper bound.