Grover's search algorithm is the optimal quantum algorithm that can search an unstructured database quadratically faster than any known classical algorithm. The role of entanglement and correlations in the search algorithm have been studied in great detail and it is known that entanglement between the qubits is necessary to gain a quadratic speedup, for pure state implementation of the Grover search algorithm. Here, we systematically investigate the behavior of quantum coherence and monogamy of entanglement in the discrete analogue of the
analog analogue of Grover search algorithm. The analog analogue of Grover search is a continuous time quantum algorithm based on the adiabatic Hamiltonian evolution that gives a quadratic speedup, similar to the original Grover search algorithm. We show that the decrease of quantum coherence, quantified using various coherence monotones, is a clear signature of attaining the maximum success probability in the analog Grover search. We also show that for any two qubit reduced density matrix of the system, the concurrence evolves in close vicinity to the increasing rate of success probability. Furthermore, we show that the system satisfies a
n-party monogamy inequality for arbitrary times, hence bounding the amount of
n-qubit entanglement during the quantum search.
This thesis explores the quantum analogues of algorithms used in mathematical optimization.The thesis focuses primarily on the iterative gradient search algorithm (algorithm for finding the minimum or maximum of a function) and the Newton-Raphson algorithm.The thesis introduces a new quantum gradient algorithm suggested by Professor Thomas Konrad and colleagues and a quantum analogue of the Newton-Raphson Method, a method for finding approximations to the roots or zeroes of a real-valued function.The quantum gradient algorithm and the quantum Newton-Raphson are shown to give a polynomial speed up over their classical analogues.