Abstract:
Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov, Ambainis, Kempe and Vazirani (2001) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speed-up in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e., walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.

Abstract:
The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally non-uniform) distribution pi(x) in time O(n) and the stationary distribution is determined by the initial state of the walk. An explicit expression for pi(x) is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p \approx 0.1$. A similar effect was previously reported for the QW on the N-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps to obtain--and preserve--a uniform distribution over the $2^n$ sites of the hypercube in the shortest possible time.

Abstract:
Using the electric and coupling approaches, we derive a series of results concerning the mixing times for the stratified random walk on the d-cube, inspired in the results of Chung and Graham (1997) Stratified random walks on the n-cube.

Abstract:
Given a finite graph G, a vertex of the lamplighter graph consists of a zero-one labeling of the vertices of G, and a marked vertex of G. For transitive graphs G, we show that, up to constants, the relaxation time for simple random walk in corresponding lamplighter graph is the maximal hitting time for simple random walk in G, while the mixing time in total variation on the lamplighter graph is the expected cover time on G. The mixing time in the uniform metric on the lamplighter graph admits a sharp threshold, and equals |G| multiplied by the relaxation time on G, up to a factor of log |G|. For the lamplighter group over the discrete two dimensional torus of sidelength n, the relaxation time is of order n^2 log n, the total variation mixing time is of order n^2 log^2 n, and the uniform mixing time is of order n^4. In dimension d>2, the relaxation time is of order n^d, the total variation mixing time is of order n^d log n, and the uniform mixing time is of order n^{d+2}. These are the first examples we know of of finite transitive graphs with uniformly bounded degrees where these three mixing time parameters are of different orders of magnitude.

Abstract:
We present an infinite family of finite planar graphs $\{X_n\}$ with degree at most five and such that for some constant $c > 0$, $$ \lambda_1(X_n) \geq c(\frac{\log \diam(X_n)}{\diam(X_n)})^2\,, $$ where $\lambda_1$ denotes the smallest non-zero eigenvalue of the graph Laplacian. This significantly simplifies a construction of Louder and Souto. We also remark that such a lower bound cannot hold when the diameter is replaced by the average squared distance: There exists a constant $c > 0$ such that for any family $\{X_n\}$ of planar graphs we have $$ \lambda_1(X_n) \leq c (\frac{1}{|X_n|^2} \sum_{x,y \in X_n} d(x,y)^2)^{-1}\,, $$ where $d$ denotes the path metric on $X_n$.

Abstract:
Mixing properties of discrete-time quantum walks on two-dimensional grids with torus-like boundary conditions are analyzed, focusing on their connection to the complexity of the corresponding abstract search algorithm. In particular, an exact expression for the stationary distribution of the coherent walk over odd-sided lattices is obtained after solving the eigenproblem for the evolution operator for this particular graph. The limiting distribution and mixing time of a quantum walk with a coin operator modified as in the abstract search algorithm are obtained numerically. On the basis of these results, the relation between the mixing time of the modified walk and the running time of the corresponding abstract search algorithm is discussed.

Abstract:
given a simple random walk on the real line , we consider the sequences of occupation times on states and associate to them martingales defined by the moments of first order of this random walk. we deduce by this way recurrent relations for the expectations of the occupation times in states before a given time , and then remarkable identities for the expectations of the absolute values of the random walk

Abstract:
Given a simple random walk on the real line , we consider the sequences of occupation times on states and associate to them martingales defined by the moments of first order of this random walk. We deduce by this way recurrent relations for the expectations of the occupation times in states before a given time , and then remarkable identities for the expectations of the absolute values of the random walk

Abstract:
In this note we establish some sufficient conditions for almost sure convergence of weighted sums of mixing sequences. These results extend and improve Theorem 3 of Chow1],Theorem 3 of Thrum2], Theorem 4.1.5 of Stout3] and Theorem 4 of Georgiev4].

Abstract:
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs lack this mixing property as well.