We show that a multistep quantum walk can be realized for a single trapped ion with an interpolation between a quantum and random walk achieved by randomizing the generalized Hadamard coin flip phase. The signature of the quantum walk is manifested not only in the ion's position but also in its phonon number, which makes an ion-trap implementation of the quantum walk feasible.

consider the random walk among n places with n(n - 1)/2 transports. we attach an exponential random variable xij to each transport between places pi and pj and take these random variables mutually independent. if transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

Quantum versions of random walks on the line and the cycle show a quadratic improvement over classical random walks in their spreading rates and mixing times respectively. Non-unitary quantum walks can provide a useful optimisation of these properties, producing a more uniform distribution on the line, and faster mixing times on the cycle. We investigate the interplay between quantum and random...

• Classical: random walk in one dimension describes a particle that moves in the positive or negative direction according to the random outcome of some unbiased binary variable (e.g., a fair coin). • Quantum: The particle with two degree of freedom (for example spin) can move left or right according to its spin and one unitary operator can play coin roles (for example Hadamard operator).In quan...

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a...

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure μn on the space of n-paths, and the μn in turn induce a quantum measure μ on the cylinder sets within the space Ω of untruncated paths. Although μ cannot be extended to a continuous quantum measure on the full σ-algebra generated by the cylinder sets, an important questi...