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Register a new userThe spectra of the unitary marix of a 2-tessellable staggered quantum walk on a graph
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Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra.
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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.
The spectrum of a semi-infinite quantum graph tube with square period cells is analyzed. The structure is obtained by rolling up a doubly periodic quantum graph into a tube along a period vector and then retaining only a semi-infinite half of the tube. The eigenfunctions associated to the spectrum of the half-tube involve all Floquet modes of the full tube. This requires solving the complex dispersion relation $D(lambda,k_1,k_2)=0$ with $(k_1,k_2)in(mathbb{C}/2pimathbb{Z})^2$ subject to the constraint $alpha k_1 + beta k_2 equiv 0$ (mod $2pi$), where $alpha$ and $beta$ are integers. The number of Floquet modes for a given $lambdainmathbb{R}$ is $2maxleft{ alpha, beta right}$. Rightward and leftward modes are determined according to an indefinite energy flux form. The spectrum may contain eigenvalues that depend on the boundary conditions, and some eigenvalues may be embedded in the continuous spectrum.
Max-plus algebra is a kind of idempotent semiring over $mathbb{R}_{max}:=mathbb{R}cup{-infty}$ with two operations $oplus := max$ and $otimes := +$.In this paper, we introduce a new model of a walk on one dimensional lattice on $mathbb{Z}$, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the $ell^2$-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk.Moreover, spectral analysis on the total time evolution operator is also given.
Birkhoffs theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group {bf E}$_{2^{2w+1}}^+$ of order $2^{2w+1}$. An associated projective group of order $2^{2w}$ equally suffices.
In a recent paper we proposed a non-Markovian random walk model with memory of the maximum distance ever reached from the starting point (home). The behavior of the walker is at variance with respect to the simple symmetric random walk (SSRW) only when she is at this maximum distance, where, having the choice to move either farther or closer, she decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold and her behavior turns out to be super-diffusive, otherwise she is timorous and her behavior turns out to be sub-diffusive. The scaling behavior vary continuously from sub-diffusive (timorous) to super-diffusive (bold) according to a single parameter $gamma in R$. We investigate here the asymptotic properties of the bold case in the non ballistic region $gamma in [0,1/2]$, a problem which was left partially unsolved in cite{S}. The exact results proved in this paper require new probabilistic tools which rely on the construction of appropriate martingales of the random walk and its hitting times.
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