Improving quantum spatial search in two dimensions

2019 ◽  
Vol 19 (7&8) ◽  
pp. 555-574
Author(s):  
Abhijith J. ◽  
Apoorva Patel
Keyword(s):  
Open Problem ◽  
Target Location ◽  
Quantum Walks ◽  
Two Dimensions ◽  
Graph Structure ◽  
Important Ingredient ◽  
Logarithmic Factor ◽  
Spatial Search ◽  
Oracle Complexity ◽  
Amplitude Amplification

The question of whether quantum spatial search in two dimensions can be made optimal has long been an open problem. We report progress towards its resolution by showing that the oracle complexity for target location can be made optimal, by increasing the number of calls to the walk operator that incorporates the graph structure by a logarithmic factor. Our algorithm does not require amplitude amplification. An important ingredient of our algorithm is the implementation of multi-step quantum walks by graph powering, using a coin space of walk-length dependent dimension, which may be of independent interest. Finally, we demonstrate how to implement quantum walks arising from powers of symmetric Markov chains using our methods.

scholarly journals Dynamic Averaging Load Balancing on Cycles

Algorithmica ◽  
2021 ◽  
Author(s):  
Dan Alistarh ◽  
Giorgi Nadiradze ◽  
Amirmojtaba Sabour
Keyword(s):  
Load Balancing ◽  
Dynamic Load ◽  
Upper Bound ◽  
Dynamic Load Balancing ◽  
Graph Structure ◽  
Logarithmic Factor ◽  
Underlying Graph ◽  
Maximum Value ◽  
Analysis Technique ◽  
The Difference

AbstractWe consider the following dynamic load-balancing process: given an underlying graph G with n nodes, in each step $$t\ge 0$$ t ≥ 0 , a random edge is chosen, one unit of load is created, and placed at one of the endpoints. In the same step, assuming that loads are arbitrarily divisible, the two nodes balance their loads by averaging them. We are interested in the expected gap between the minimum and maximum loads at nodes as the process progresses, and its dependence on n and on the graph structure. Peres et al. (Random Struct Algorithms 47(4):760–775, 2015) studied the variant of this process, where the unit of load is placed in the least loaded endpoint of the chosen edge, and the averaging is not performed. In the case of dynamic load balancing on the cycle of length n the only known upper bound on the expected gap is of order $$\mathcal {O}( n \log n )$$ O ( n log n ) , following from the majorization argument due to the same work. In this paper, we leverage the power of averaging and provide an improved upper bound of $$\mathcal {O} ( \sqrt{n} \log n )$$ O ( n log n ) . We introduce a new potential analysis technique, which enables us to bound the difference in load between k-hop neighbors on the cycle, for any $$k \le n/2$$ k ≤ n / 2 . We complement this with a “gap covering” argument, which bounds the maximum value of the gap by bounding its value across all possible subsets of a certain structure, and recursively bounding the gaps within each subset. We also show that our analysis can be extended to the specific instance of Harary graphs. On the other hand, we prove that the expected second moment of the gap is lower bounded by $$\Omega (n)$$ Ω ( n ) . Additionally, we provide experimental evidence that our upper bound on the gap is tight up to a logarithmic factor.

scholarly journals Optimality of spatial search via continuous-time quantum walks

2020 ◽  
Vol 102 (3) ◽  
Author(s):  
Shantanav Chakraborty ◽  
Leonardo Novo ◽  
Jérémie Roland
Keyword(s):  
Continuous Time ◽  
Quantum Walks ◽  
Spatial Search ◽  
Time Quantum

scholarly journals Electric quantum walks in two dimensions

2016 ◽  
Vol 93 (3) ◽  
Author(s):  
Luis A. Bru ◽  
Margarida Hinarejos ◽  
Fernando Silva ◽  
Germán J. de Valcárcel ◽  
Eugenio Roldán
Keyword(s):  
Quantum Walks ◽  
Two Dimensions

scholarly journals Dirac Spatial Search with Electric Fields

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1441
Author(s):  
Julien Zylberman ◽  
Fabrice Debbasch
Keyword(s):  
Dirac Equation ◽  
Electric Field ◽  
Time Scales ◽  
Electric Fields ◽  
Point Charge ◽  
Quantum Walks ◽  
Spatial Search ◽  
Speed Up ◽  
Finite Period ◽  
Non Local

Electric Dirac quantum walks, which are a discretisation of the Dirac equation for a spinor coupled to an electric field, are revisited in order to perform spatial searches. The Coulomb electric field of a point charge is used as a non local oracle to perform a spatial search on a 2D grid of N points. As other quantum walks proposed for spatial search, these walks localise partially on the charge after a finite period of time. However, contrary to other walks, this localisation time scales as N for small values of N and tends asymptotically to a constant for larger Ns, thus offering a speed-up over conventional methods.

Shape Retrieval and Classification Based on Geodesic Paths in Skeleton Graphs

2012 ◽  
pp. 190-215
Author(s):  
Xiang Bai ◽  
Chunyuan Li ◽  
Xingwei Yang ◽  
Longin Jan Latecki
Keyword(s):  
Open Problem ◽  
Graph Matching ◽  
Shape Retrieval ◽  
Shape Similarity ◽  
Shape Classification ◽  
Graph Structure ◽  
Topological Graph ◽  
Direct Graph ◽  
Skeleton Graph ◽  
Geodesic Paths

Skeleton- is well-known to be superior to contour-based representation when shapes have large nonlinear variability, especially articulation. However, approaches to shape similarity based on skeletons suffer from the instability of skeletons, and matching of skeleton graphs is still an open problem. To deal with this problem for shape retrieval, the authors first propose to match skeleton graphs by comparing the geodesic paths between skeleton endpoints. In contrast to typical tree or graph matching methods, they do not explicitly consider the topological graph structure. Their approach is motivated by the fact that visually similar skeleton graphs may have completely different topological structures, while the paths between their end nodes still remain similar. The proposed comparison of geodesic paths between endpoints of skeleton graphs yields correct matching results in such cases. The experimental results demonstrate that the method is able to produce correct results in the presence of articulations, stretching, and contour deformations. The authors also utilize the geodesic skeleton paths for shape classification. Similar to shape retrieval, direct graph matching algorithms like graph edit distance have great difficulties with the instability of the skeleton graph structure. In contrast, the representation based on skeleton paths remains stable. Therefore, a simple Bayesian classifier is able to obtain excellent shape classification results.

scholarly journals Non-Markovianity is not a resource for quantum spatial search on a star graph subject to generalized percolation

2018 ◽  
Vol 5 (1) ◽  
pp. 40-49 ◽  
Author(s):  
Matteo A. C. Rossi ◽  
Marco Cattaneo ◽  
Matteo G. A. Paris ◽  
Sabrina Maniscalco
Keyword(s):  
Continuous Time ◽  
Quantum Algorithm ◽  
Quantum Walks ◽  
Quantum Search ◽  
Star Graph ◽  
Correlated Noise ◽  
Random Telegraph Noise ◽  
Spatial Search ◽  
Telegraph Noise ◽  
Time Quantum

Abstract Continuous-time quantum walks may be exploited to enhance spatial search, i.e., for finding a marked element in a database structured as a complex network. However, in practical implementations, the environmental noise has detrimental effects, and a question arises on whether noise engineering may be helpful in mitigating those effects on the performance of the quantum algorithm. Here we study whether time-correlated noise inducing non-Markovianity may represent a resource for quantum search. In particular, we consider quantum search on a star graph, which has been proven to be optimal in the noiseless case, and analyze the effects of independent random telegraph noise (RTN) disturbing each link of the graph. Upon exploiting an exact code for the noisy dynamics, we evaluate the quantum non-Markovianity of the evolution, and show that it cannot be considered as a resource for this algorithm, since its presence is correlated with lower probabilities of success of the search.

Oculomotor Tracking in Two Dimensions

1999 ◽  
Vol 81 (4) ◽  
pp. 1597-1602 ◽  
Author(s):  
Kevin C. Engel ◽  
John H. Anderson ◽  
John F. Soechting
Keyword(s):  
Smooth Pursuit ◽  
Target Location ◽  
Two Dimensions ◽  
Positional Error ◽  
Additional Insight ◽  
Saccade Onset ◽  
Straight Line ◽  
Average Slope ◽  
Pursuit Movement ◽  
Pursuit Velocity

Oculomotor tracking in two dimensions. Results from studies of oculomotor tracking in one dimension have indicated that saccades are driven primarily by errors in position, whereas smooth pursuit movements are driven primarily by errors in velocity. To test whether this result generalizes to two-dimensional tracking, we asked subjects to track a target that moved initially in a straight line then changed direction. We found that the general premise does indeed hold true; however, the study of oculomotor tracking in two dimensions provides additional insight. The first saccade was directed slightly in advance of target location at saccade onset. Thus its direction was related primarily to angular positional error. The direction of the smooth pursuit movement after the saccade was related linearly to the direction of target motion with an average slope of 0.8. Furthermore the magnitude and direction of smooth pursuit velocity did not change abruptly; consequently the direction of smooth pursuit appeared to rotate smoothly over time.

scholarly journals Bounds on integrals with respect to multivariate copulas

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Michael Preischl
Keyword(s):  
Lower Bounds ◽  
Open Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Multidimensional Case ◽  
Assignment Problems ◽  
Dimensional Dependence ◽  
Dependence Measures

AbstractIn this paper, we present a method to obtain upper and lower bounds on integrals with respect to copulas by solving the corresponding assignment problems (AP’s). In their 2014 paper, Hofer and Iacó proposed this approach for two dimensions and stated the generalization to arbitrary dimensons as an open problem. We will clarify the connection between copulas and AP’s and thus find an extension to the multidimensional case. Furthermore, we provide convergence statements and, as applications, we consider three dimensional dependence measures as well as an example from finance.

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