Laplacian versus adjacency matrix in quantum walk search
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This work algorithmically explores search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix, and finds that the two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked Vertices, and in what constitutes a natural initial state.Abstract:
A quantum particle evolving by Schrodinger's equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace's operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.read more
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Optimality of spatial search via continuous-time quantum walks
TL;DR: This work derives general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled and shows the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitions into clusters.
Journal ArticleDOI
Quantum spatial search on graphs subject to dynamical noise
TL;DR: In this paper, quantum spatial search on graphs and its implementation by continuous-time quantum walks in the presence of dynamical noise was studied and it was shown that noiseless spatial search shows optimal quantum speedup in the latter, in the computational limit $N\ensuremath{\gg}1$.
Journal ArticleDOI
Quantum walk search on the complete bipartite graph
Mason L. Rhodes,Thomas G. Wong +1 more
TL;DR: In this article, the authors investigated how Grover's algorithm can evolve with respect to the number of marked and unmarked vertices in each partite set, showing that the success probability can vary greatly from one time step to the next, even alternating between 0 and 1.
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Vertices cannot be hidden from quantum spatial search for almost all random graphs
TL;DR: In this article, the authors show that all nodes can be found optimally for almost all random Erdős-Renyi (n,p) graphs using continuous-time quantum spatial search procedure.
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Quantum Walk Search on Johnson Graphs
TL;DR: In this paper, it was shown that a change of basis is needed for degenerate perturbation theory to accurately describe the dynamics of Johnson graphs with fixed k-element subsets of the symbols.
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