2,954 research outputs found
A remark on zeta functions of finite graphs via quantum walks
From the viewpoint of quantum walks, the Ihara zeta function of a finite
graph can be said to be closely related to its evolution matrix. In this note
we introduce another kind of zeta function of a graph, which is closely related
to, as to say, the square of the evolution matrix of a quantum walk. Then we
give to such a function two types of determinant expressions and derive from it
some geometric properties of a finite graph. As an application, we illustrate
the distribution of poles of this function comparing with those of the usual
Ihara zeta function.Comment: 14 pages, 1 figur
Continuous-time quantum walk on integer lattices and homogeneous trees
This paper is concerned with the continuous-time quantum walk on Z, Z^d, and
infinite homogeneous trees. By using the generating function method, we compute
the limit of the average probability distribution for the general isotropic
walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous
trees. In addition, we compute the asymptotic approximation for the probability
of the return to zero at time t in all these cases.Comment: The journal version (save for formatting); 19 page
Wigner formula of rotation matrices and quantum walks
Quantization of a random-walk model is performed by giving a qudit (a
multi-component wave function) to a walker at site and by introducing a quantum
coin, which is a matrix representation of a unitary transformation. In quantum
walks, the qudit of walker is mixed according to the quantum coin at each time
step, when the walker hops to other sites. As special cases of the quantum
walks driven by high-dimensional quantum coins generally studied by Brun,
Carteret, and Ambainis, we study the models obtained by choosing rotation as
the unitary transformation, whose matrix representations determine quantum
coins. We show that Wigner's -dimensional unitary representations of
rotations with half-integers 's are useful to analyze the probability laws
of quantum walks. For any value of half-integer , convergence of all moments
of walker's pseudovelocity in the long-time limit is proved. It is generally
shown for the present models that, if is even, the probability measure
of limit distribution is given by a superposition of terms of scaled
Konno's density functions, and if is odd, it is a superposition of
terms of scaled Konno's density functions and a Dirac's delta function at the
origin. For the two-, three-, and four-component models, the probability
densities of limit distributions are explicitly calculated and their dependence
on the parameters of quantum coins and on the initial qudit of walker is
completely determined. Comparison with computer simulation results is also
shown.Comment: v2: REVTeX4, 15 pages, 4 figure
Absorption problems for quantum walks in one dimension
This paper treats absorption problems for the one-dimensional quantum walk
determined by a 2 times 2 unitary matrix U on a state space {0,1,...,N} where N
is finite or infinite by using a new path integral approach based on an
orthonormal basis P, Q, R and S of the vector space of complex 2 times 2
matrices. Our method studied here is a natural extension of the approach in the
classical random walk.Comment: 15 pages, small corrections, journal reference adde
The Vertex-Face Correspondence and the Elliptic 6j-symbols
A new formula connecting the elliptic -symbols and the fusion of the
vertex-face intertwining vectors is given. This is based on the identification
of the fusion intertwining vectors with the change of base matrix elements
from Sklyanin's standard base to Rosengren's natural base in the space of even
theta functions of order . The new formula allows us to derive various
properties of the elliptic -symbols, such as the addition formula, the
biorthogonality property, the fusion formula and the Yang-Baxter relation. We
also discuss a connection with the Sklyanin algebra based on the factorised
formula for the -operator.Comment: 23 page
Invariant Measures and Decay of Correlations for a Class of Ergodic Probabilistic Cellular Automata
We give new sufficient ergodicity conditions for two-state probabilistic
cellular automata (PCA) of any dimension and any radius. The proof of this
result is based on an extended version of the duality concept. Under these
assumptions, in the one dimensional case, we study some properties of the
unique invariant measure and show that it is shift-mixing. Also, the decay of
correlation is studied in detail. In this sense, the extended concept of
duality gives exponential decay of correlation and allows to compute
explicitily all the constants involved
Survival probability of the Grover walk on the ladder graph
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 Mare\v
s et al., Phys. Rev. A 101, 032113 (2020), 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 , 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 can change from increasing and converging exponentially quickly
to decreasing and converging like 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
Stability conditions and positivity of invariants of fibrations
We study three methods that prove the positivity of a natural numerical
invariant associated to parameter families of polarized varieties. All
these methods involve different stability conditions. In dimension 2 we prove
that there is a natural connection between them, related to a yet another
stability condition, the linear stability. Finally we make some speculations
and prove new results in higher dimension.Comment: Final version, to appear in the Springer volume dedicated to Klaus
Hulek on the occasion of his 60-th birthda
Recurrence of biased quantum walks on a line
The Polya number of a classical random walk on a regular lattice is known to
depend solely on the dimension of the lattice. For one and two dimensions it
equals one, meaning unit probability to return to the origin. This result is
extremely sensitive to the directional symmetry, any deviation from the equal
probability to travel in each direction results in a change of the character of
the walk from recurrent to transient. Applying our definition of the Polya
number to quantum walks on a line we show that the recurrence character of
quantum walks is more stable against bias. We determine the range of parameters
for which biased quantum walks remain recurrent. We find that there exist
genuine biased quantum walks which are recurrent.Comment: Journal reference added, minor corrections in the tex
Elliptic Deformed Superalgebra
We introduce the elliptic superalgebra as one
parameter deformation of the quantum superalgebra . For an
arbitrary level we give the bosonization of the elliptic
superalgebra and the screening currents that commute
with modulo total difference.Comment: LaTEX, 25 page
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