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.
Decoherence in quantum searches, and in the Grover search, in particular, has already been extensively studied, leading very quickly to the loss of the quadratic speedup over the classical case, when searching for some target (marked) element within a set of size N. The noise models used were, however, almost always global. In this paper, we study Grover search under the influence of localized partially dephasing noise of rate p. We find that, in the case when the size k of the affected subspace is much smaller than N and the target is unaffected by the noise, namely when $kp\ensuremath{\ll}\sqrt{N}$, the quadratic speedup is retained. Once these restrictions are not met, the quadratic speedup is lost. If the target is affected by the noise, the noise rate needs to scale as 1/N to keep the speedup. We also observe an intermediate region, where if $k\ensuremath{\sim}{N}^{\ensuremath{\mu}}$ and the target is unaffected, the speedup seems to obey ${N}^{\ensuremath{\mu}}$, which for $\ensuremath{\mu}>0.5$ is worse than the quantum, but better than the classical case. We also put obtained results for quantum searches into perspective of quantum walks and searches on graphs.
Abstract. Traditional tree search algorithms supply a blueprint for modeling prob-lem solving behaviour. A diverse spectrum of problems can be formulated in terms of tree search. Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. In this work we consider the impact of incorporating classical search con-cepts alongside Grover’s algorithm into a hybrid quantum search system. Some of the crucial points examined include: (1) the reverberations of contemplating the use of non-constant branching factors; (2) determining the consequences of incorporating an heuristic perspective into a quantum tree search model.