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Register a new userThe parallel Grover as dynamic system
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Alexander Goscinski
Publication date
2018
fields
Physics
and research's language is
English
Authors
Alexander Goscinski
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A sequential application of the Grover algorithm to solve the iterated search problem has been improved by Ozhigov by parallelizing the application of the oracle. In this work a representation of the parallel Grover as dynamic system of inversion about the mean and Grover operators is given. Within this representation the parallel Grover for $k = 2$ can be interpreted as rotation in three-dimensional space and it can be shown that the sole application of the parallel Grover operator does not lead to a solution for $k > 2$. We propose a solution for $k = 3$ with a number of approximately $1.51sqrt{N}$ iterations.
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We provide first evidence that under certain conditions, 1/2-spin fermions may naturally behave like a Grover search, looking for topological defects in a material. The theoretical framework is that of discrete-time quantum walks (QW), i.e. local unitary matrices that drive the evolution of a single particle on the lattice. Some QW are well-known to recover the $(2+1)$--dimensional Dirac equation in continuum limit, i.e. the free propagation of the 1/2-spin fermion. We study two such Dirac QW, one on the square grid and the other on a triangular grid reminiscent of graphene-like materials. The numerical simulations show that the walker localises around the defects in $O(sqrt{N})$ steps with probability $O(1/log{N})$, in line with previous QW search on the grid. The main advantage brought by those of this paper is that they could be implemented as `naturally occurring freely propagating particles over a surface featuring topological---without the need for a specific oracle step. From a quantum computing perspective, however, this hints at novel applications of QW search : instead of using them to look for `good solutions within the configuration space of a problem, we could use them to look for topological properties of the entire configuration space.
The Grover walk is one of well-studied quantum walks on graphs and its periodicity is investigated to reveal the relation between the quantum walk and the underlying graph. Especially, characterization of graphs exhibiting a periodic Grover walk is intensively studied. Yoshie has already characterized such graphs having a periodic Grover walk with periods $2, 3, 4$ and $5$. In the work, it is expected that the graphs exhibiting a periodic Grover walk with odd period are the cycles with odd length. In this paper, we address that problem and obtained the perfect answer, that is, we perfectly characterize the class of graphs exhibiting an odd-periodic Grover walk by a combinatorial method. More precisely, we solve the problem by analyzing the characteristic polynomial of a weighted adjacency matrix of the graph.
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.
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.
Recently the Ihara zeta function for the finite graph was extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno-Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and the zeta function based on the Konno-Sato theorem Grover/Zeta Correspondence here.
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