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We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an $ell^infty$-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 proportion 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 $(r+1)$.
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The principles of classical mechanics have shown that the inertial quality of mass is characterized by the kinetic energy. This, in turn, establishes the connection between geometry and mechanics. We aim to exploit such a fundamental principle for information geometry entering the realm of mechanics. According to the modification of curve energy stated by Amari and Nagaoka for a smooth manifold $mathrm{M}$ endowed with a dual structure $(mathrm{g}, abla, abla^*)$, we consider $ abla$ and $ abla^*$ kinetic energies. Then, we prove that a recently introduced canonical divergence and its dual function coincide with Hamilton principal functions associated with suitable Lagrangian functions when $(mathrm{M},mathrm{g}, abla, abla^*)$ is dually flat. Corresponding dynamical systems are studied and the tangent dynamics is outlined in terms of the Riemannian gradient of the canonical divergence. Solutions of such dynamics are proved to be $ abla$ and $ abla^*$ geodesics connecting any two points sufficiently close to each other. Application to the standard Gaussian model is also investigated.
Given two Hilbert spaces, $mathcal{H}$ and $mathcal{K}$, we introduce an abstract unitary operator $U$ on $mathcal{H}$ and its discriminant $T$ on $mathcal{K}$ induced by a coisometry from $mathcal{H}$ to $mathcal{K}$ and a unitary involution on $mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpinski lattice, which is pure point and has a Cantor-like structure.
We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the $ell^infty$-category initial state so that the internal graph receives time independent input from the tails, say $boldsymbol{alpha}_{in}$, at every time step. We show that the response of the Szegedy walk to the input, which is the output, say $boldsymbol{beta}_{out}$, from the internal graph to the tails in the long time limit, is drastically changed depending on the reversibility of the underlying random walk. If the underlying random walk is reversible, we have $boldsymbol{beta}_{out}=mathrm{Sz}(boldsymbol{m}_{delta E})boldsymbol{alpha}_{in}$, where the unitary matrix $mathrm{Sz}(boldsymbol{m}_{delta E})$ is the reflection matrix to the unit vector $boldsymbol{m}_{delta E}$ which is determined by the boundary of the internal graph $delta E$. Then the global dynamics so that the internal graph is regarded as one vertex recovers the local dynamics of the Szegedy walk in the long time limit. Moreover if the underlying random walk of the Szegedy walk is reversible, then we obtain that the stationary state is expressed by a linear combination of the reversible measure and the electric current on the electric circuit determined by the internal graph and the random walks reversible measure. On the other hand, if the underlying random walk is not reversible, then the unitary matrix is just a phase flip; that is, $boldsymbol{beta}_{out}=-boldsymbol{alpha}_{in}$, and the stationary state is similar to the current flow but satisfies a different type of the Kirchhoff laws.
We consider the discrete-time quantum walk whose local dynamics is denoted by $C$ at the perturbed region ${0,1,dots,M-1}$ and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow $omega^n$ at time $n$ $(|omega|=1)$. From this expression, we compute the scattering on the surface of $-1$ and $M$ and also compute the quantity how quantum walker accumulates in the perturbed region; namely the energy of the quantum walk, in the long time limit. We find a discontinuity of the energy with respect to the frequency of the inflow.
This paper studies the spectrum of a multi-dimensional split-step quantum walk with a defect that cannot be analysed in the previous papers. To this end, we have developed a new technique which allow us to use a spectral mapping theorem for the one-defect model. We also derive the time-averaged limit measure for one-dimensional case as an application of the spectral analysis.
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