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 Physics , 2007, DOI: 10.1103/PhysRevA.77.042312 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.
 Ronen Eldan Mathematics , 2011, Abstract: We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.  Physics , 2010, Abstract: In this paper, we study mixing and large decoherence in continuous-time quantum walks on one dimensional regular networks, which are constructed by connecting each node to its$2l$nearest neighbors($l\$ on either side). In our investigation, the nodes of network are represented by a set of identical tunnel-coupled quantum dots in which decoherence is induced by continuous monitoring of each quantum dot with nearby point contact detector. To formulate the decoherent CTQWs, we use Gurvitz model and then calculate probability distribution and the bounds of instantaneous and average mixing times. We show that the mixing times are linearly proportional to the decoherence rate. Moreover, adding links to cycle network, in appearance of large decoherence, decreases the mixing times.