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Register a new userCoherence depletion in the Grover quantum search algorithm
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We investigate the role of quantum coherence depletion (QCD) in Grover search algorithm (GA) by using several typical measures of quantum coherence and quantum correlations. By using the relative entropy of coherence measure ($mathcal{C}_r$), we show that the success probability depends on the QCD. The same phenomenon is also found by using the $l_1$ norm of coherence measure ($mathcal{C}_{l_1}$). In the limit case, the cost performance is defined to characterize the behavior about QCD in enhancing the success probability of GA, which is only related to the number of searcher items and the scale of database, no matter using $mathcal{C}_r$ or $mathcal{C}_{l_1}$. In generalized Grover search algorithm (GGA), the QCD for a class of states increases with the required optimal measurement time. In comparison, the quantification of other quantum correlations in GA, such as pairwise entanglement, multipartite entanglement, pairwise discord and genuine multipartite discord, cannot be directly related to the success probability or the optimal measurement time. Additionally, we do not detect pairwise nonlocality or genuine tripartite nonlocality in GA since Clauser-Horne-Shimony-Holt inequality and Svetlichnys inequality are not violated.
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In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We further show that the final state can be expressed in terms of a single oracle operator of the Grover-search and phase-rotation factors. We discuss how the additive form can be utilized so that it effectively reduces the computational load of the iterative search, and we propose an effective shortcut gate that realizes the same outcome as the iterative search.
We consider the Grover search algorithm implementation for a quantum register of size $N = 2^k$ using k (or k +1) microwave- and laser-driven Rydberg-blockaded atoms, following the proposal by M{o}lmer, Isenhower, and Saffman [J. Phys. B 44, 184016 (2011)]. We suggest some simplifications for the microwave and laser couplings, and analyze the performance of the algorithm for up to k = 4 multilevel atoms under realistic experimental conditions using quantum stochastic (Monte-Carlo) wavefunction simulations.
Grovers algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N-r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T sim O(sqrt{N/r}) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial amplitude distributions of the marked or unmarked states.
Phase matching has been studied for the Grover algorithm as a way of enhancing the efficiency of the quantum search. Recently Li and Li found that a particular form of phase matching yields, with a single Grover operation, a success probability greater than 25/27 for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than 1/3. Although this single operation eliminates the oscillations of the success probability that occur with multiple Grover operations, the latter oscillations reappear with multiple iterations of Li and Lis phase matching. In this paper we introduce a multi-phase matching subject to a certain matching rule by which we can obtain a multiple Grover operation that with only a few iterations yields a success probability that is almost constant and unity over a wide range of the fraction of marked items. As an example we show that a multi-phase operation with six iterations yields a success probability between 99.8% and 100% for a fraction of marked states of 1/10 or larger.
Quantum search is a quantum mechanical technique for searching N possibilities in only sqrt(N) steps. This has been proved to be the best possible algorithm for the exhuastive search problem in the sense the number of queries it requires cannot be reduced. However, as this paper shows, the number of non-query operations, and thus the total number of operations, can be reduced. The number of non-query unitary operations can be reduced by a factor of log N/alpha*log(log N) while increasing the number of queries by a factor of only (1+(log N)^{-alpha}). Various choices of alpha yield different variants of the algorithm. For example, by choosing alpha to be O(log N/log(log N)), the number of non-query unitary operations can be reduced by 40% while increasing the number of queries by just two.
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