Entanglement in the Quantum Search Algorithm

Abstract

Quantum mechanics has fascinated researchers for long. Its weird and strange world offers limitless boundaries and finds varied applications. As transistors are presumed to shrink to the size of an atom, quantum mechanics is expected to dominate computing in the coming decades. Laws of quantum mechanics have shown to make computing incredibly fast much faster than the best known classical computer. Naturally, quantum algorithms have been proven to have a considerable speed up over their classical counterparts. However, the precise reason behind this speed up is yet under much debate. Entanglement, arguably the strangest of all quantum phenomena is suspected to be the central reason behind this. However, the lack of a proper mathematical structure for entanglement for higher qubits has made its study even more difficult. This thesis analyses the nature of entanglement in the quantum search algorithm, also known as the Grover's search algorithm. This algorithm enables the searching of an element from an unstructured database, quadratically faster than the best known classical algorithm. Clearly, entanglement has a role in this and hence its study is of utmost importance. The thesis reveals the role played by entanglement in the Grover's search algorithm, first for two qubits and then for multiqubit scenarios. For two qubits, concurrence has been used to quantify the entanglement and the effect of noise causing decoherence has also been taken into account. It has been shown that under the influence of noise (namely amplitude damping), there exists non-classical correlations beyond entanglement by making use of quantum discord. For n-qubits, geometric measure of entanglement has been used to precisely evaluate the entanglement at each iteration of the algorithm. The variation of entanglement with the increase in the number of qubits and also with the number of marked or solution states is revealed. Numerically, it is seen that the behaviour of the maximum value of entanglement is monotonous with the number of qubits. Also, for a given value of the number of qubits, the change in the marked states alter the amount of entanglement. The amount of entanglement in the final state of the algorithm has been shown to depend solely on the nature of the marked states. Explicit analytical expressions are given showing the variation of entanglement with the number of iterations and the global maximum value of entanglement attained across all iterations of the algorithm.