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arXiv:2201.01926 · Journal of Physics A Mathematical and Theoretical

A comfortable graph structure for Grover walk

Yusuke Higuchi, Mohamed Sabri, Etsuo Segawa

Published May 19, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/acd735

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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.
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 .
We consider the stationary state of a quantum walk on the finite path, where the sink and source are set at the left and right boundaries. The quantum coin is uniformly placed at every vertex of the path graph. At every time step, a new quantum walker penetrates into the internal from the left boundary and also some existing quantum walkers in the internal goes out to the sinks located in the left and right boundaries. The square modulus of the stationary state at each vertex is regarded as the comfortability for a quantum walker to this vertex in this paper. We show the weak convergence theorem for the scaled limit distribution of the comfortability in the limit of the length of the path.
We consider a quantum walk model on a finite graph which has an interaction with the outside. Here a quantum walker from the outside penetrates the graph and also a quantum walker in the graph goes out to the outside at every time step. This dynamics of the quantum walk converges to a stationary state. In this paper, we estimate the speed of the convergence to the stationary state on the κκκ-regular graph with the uniformly inserting of the inflow to the graph. We show that larger degree of the regular graph makes the convergence speed of this quantum walk model slower.
We consider a quantum walk model on a finite graph which has an interaction with the outside. Here a quantum walker from the outside penetrates the graph and also a quantum walker in the graph goes out to the outside at every time step. This dynamics of the quantum walk converges to a stationary state. In this paper, we estimate the speed of the convergence to the stationary state on the κ-regular graph with the uniform insertion of the inflow to the graph. We show that larger degree of the regular graph makes the convergence time of this quantum walk model slower under this set-up.
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.
We discuss a particular kind of quantum walk on a general graph. We affix two semi-infinite lines to a general finite graph, which we call tails. On the tails, the particle making the walk simply advances one unit at each time step, so that its behavior there is analogous to free propagation We are interested in how many steps it will take the particle, starting on one tail and propagating through the graph (where its propagation is not free), to emerge onto the other tail. The probability to make such a walk in n steps and the hitting time for such a walk can be expressed in terms of the transmission amplitude for the graph, which is one element of its S matrix. Demonstrating this necessitates a study of the analyticity properties of the transmission and reflection amplitudes of a graph. We show that the graph can have bound states that cannot be accessed by a particle entering the graph from one of its tails. Time-reversal invariance of a quantum walk is defined and used to show that the transmission amplitudes for the particle entering the graph from different directions are the same if the walk is time-reversal invariant.