Non-classical Correlations in the Quantum Search Algorithm

Abstract: Entanglement lies at the heart of quantum mechanics and has no classical analogue. It is central to the speed up achieved by quantum algorithms over their classical counterparts. The Grover's search algorithm is one such algorithm which enables us to achieve a quadratic speed up over any known classical algorithm that searches for an element in an unstructured database. Here, we analyse and quantify the effects of entanglement in the generalized version of this algorithm for two qubits. By 'generalized', it is meant that the use of any arbitrary single qubit unitary gate is permitted to create superposed states. Our analysis has been firstly on a noise free environment and secondly in the presence of noise. In the absence of noise, we establish a relation between the concurrence and the amplitude of the final state thereby showing the explicit effects of entanglement on the same. Moreover, the effects of noisy channels, namely amplitude and phase damping channels are studied. We investigate the amount of quantum correlation in the states obtained after the phase inversion stage of the algorithm followed by interaction of those states with the noisy environment. The quantum correlations are quantified by geometric discord. It has been revealed that the states generated after the effect of amplitude damping on the phase inverted states of the quantum search algorithm possess non-zero quantum correlation even when entanglement is absent. However, this is absent in the phase damping scenario.