arXiv:2105.03111 · Symmetry

Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs

Norio Konno, Etsuo Segawa, M. Štefaňák

Published Jun 29, 2021• 5 citations

Abstract

We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in graph theory.

DOI: 10.3390/sym13071169

Related papers

A dynamical system induced by quantum walk

Yusuke Higuchi, Etsuo Segawa

Abstract We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an -infinite external source from one of the tails as the initial state. We show that for any connected internal graph, a stationary state exists, moreover a perfect transmission to the opposite tail always occurs in the long time limit. We also show that the lower bound of the norm of the stationary measure restricted to the internal graph is proportional to the number of edges of this graph. Furthermore when we add more tails (e.g. r -tails) to the internal graph, then we find that from the temporal and spatial global view point, the scattering to each tail in the long time limit coincides with the local one-step scattering manner of the Grover walk at a vertex whose degree is .

arXiv:1812.04730Journal of Physics A Mathematical and TheoreticalJul 30, 201926 citations

Survival probability of the Grover walk on the ladder graph

Etsuo Segawa, Shigeyuki Koyama, Norio Konno +1 more

Abstract We provide a detailed analysis of the survival probability of the Grover walk on the ladder graph with an absorbing sink. This model was discussed in Mareš et al (2020 Phys. Rev. A 101 032113 ), as an example of counter-intuitive behaviour in quantum transport where it was found that the survival probability decreases with the length of the ladder L , despite the fact that the number of dark states increases. An orthonormal basis in the dark subspace is constructed, which allows us to derive a closed formula for the survival probability. It is shown that the course of the survival probability as a function of L can change from increasing and converging exponentially quickly to decreasing and converging like L −1 simply by attaching a loop to one of the corners of the ladder. The interplay between the initial state and the graph configuration is investigated.

arXiv:2205.13188Journal of Physics A Mathematical and TheoreticalApr 24, 20232 citations

A comfortable graph structure for Grover walk

Yusuke Higuchi, Mohamed Sabri, Etsuo Segawa

Abstract We consider a Grover walk model on a finite internal graph, which is connected with a finite number of semi-infinite length paths and receives the alternative inflows along these paths at each time step. After the long time scale, we know that the behavior of such a Grover walk should be stable, that is, this model has a stationary state. In this paper our objectives are to give some characterization upon the scattering of the stationary state on the surface of the internal graph and upon the energy of this state in the interior. For the scattering, we concretely give a scattering matrix, whose form is changed depending on whether the internal graph is bipartite or not. On the other hand, we introduce a comfortability function of a graph for the quantum walk, which shows how many quantum walkers can stay in the interior, and we succeed in showing the comfortability of the walker in terms of combinatorial properties of the internal graph.

arXiv:2201.01926Journal of Physics A Mathematical and TheoreticalMay 19, 20231 citation

Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

J. Mareš, J. Novotný, M. Štefaňák +1 more

Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping---there can be nonzero probability of the quantum walker being localized in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned nonzero amplitudes at each step. We study quantum walks with the flip-flop shift operator and the Grover coin, where this effect has been identified previously. For the version of the walk further modified by a random dynamical disruption of the graph (percolated quantum walks) we provide a recipe for the construction of a complete basis of the subspace of trapped states allowing to determine the asymptotic probability of trapping for arbitrary finite connected simple graphs, thus significantly generalizing the previously known result restricted to planar 3-regular graphs. We show how the position of the source and sink together with the graph geometry and its modifications affect the excitation transport. This gives us a deep insight into processes where elongation or addition of dead-end subgraphs may surprisingly result in enhanced transport and we design graphs exhibiting this pronounced behavior. In some cases this even provides closed-form formulas for the asymptotic transport probability in dependence on some structure parameters of the graphs.

arXiv:2202.09582Physical Review AJun 10, 20222 citations

Quantum walks induced by Dirichlet random walks on infinite trees

Yusuke Higuchi, Etsuo Segawa

Abstract We consider the Grover walk on infinite trees from the viewpoint of spectral analysis. From the previous work, infinite regular trees provide localization. In this paper, we give the complete characterization of the eigenspace of this Grover walk, which involves localization of its behavior and recovers the previous work. Our result suggests that the Grover walk on infinite trees may be regarded as a limit of the quantum walk induced by the isotropic random walk with the Dirichlet boundary condition at the n -th depth rather than one with the Neumann boundary condition.

arXiv:1703.01334Journal of Physics A Mathematical and TheoreticalJan 19, 201817 citations

Percolated quantum walks with a general shift operator: From trapping to transport

Jan Mareš, Jaroslav Novotný, Igor Jex

We present an alternative definition of discrete-time coined quantum walks convenient for capturing a rather broad spectrum of a walker's behavior on arbitrary graphs. It includes and covers both the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find that for these walks with cyclic shift operators, the existence of an edge-3 coloring of the graph allows for nonstationary asymptotic behavior of the walk. For different shift operators, the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory, we treat a single-excitation transport on a percolated cube.

arXiv:1812.02519Physical Review AApr 30, 201912 citations