Related papers: Random walk questions for linear quantum groups
We investigate quantum walks in multiple dimensions with different quantum coins. We augment the model by assuming that at each step the amplitudes of the coin state are multiplied by random phases. This model enables us to study in detail…
Quantum walks have emerged as an interesting alternative to the usual circuit model for quantum computing. While still universal for quantum computing, the quantum walk model has very different physical requirements, which lends itself more…
Quantum random walks are constructed on operator spaces with the aid of matrix-space lifting, a type of ampliation intermediate between those provided by spatial and ultraweak tensor products. Using a form of Wiener-Ito decomposition, a…
We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure.…
We use discrete-event simulation on a digital computer to study two different models of experimentally realizable quantum walks. The simulation models comply with Einstein locality, are as "realistic" as the one of the simple random walk in…
Initial steps are presented towards understanding which finitely generated groups are almost surely generated as semigroups by the path of a random walk on the group.
We study whether the probability distribution of a discrete quantum walk can get arbitrarily close to uniform, given that the walk starts with a uniform superposition of the outgoing arcs of some vertex. We establish a characterization of…
Given a discrete quantum group $H$ with a finite normal quantum subgroup $G$, we show that any positive, possibly unbounded, harmonic function on $H$ with respect to an irreducible invariant random walk is $G$-invariant. This implies that,…
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise quantum walks have shown much potential as a frame- work for developing new quantum algorithms.…
Let $\Gamma$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $\mu$ a probability measure on $\Gamma$ whose support generates a non-elementary subsemigroup. Under the assumption that $\mu$ has a finite…
For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant…
Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata…
We consider a discrete-time quantum walk $W_{t,\kappa}$ at time $t$ on a graph with joined half lines $\mathbb{J}_\kappa$, which is composed of $\kappa$ half lines with the same origin. Our analysis is based on a reduction of the walk on a…
A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac quantum walks, provide a description of the free evolution of relativistic quantum…
We present an experimental implementation of the coined discrete time quantum walk on a square using a three qubit liquid state nuclear magnetic resonance (NMR) quantum information processor (QIP). Contrary to its classical counterpart, we…
Quantum random walks use interference to obtain faster state space exploration, which can be used for algorithmic purposes. Photonic technologies provide a natural platform for many recent experimental demonstrations. Here we analyze…
We study fine structure related to finitely supported random walks on infinite finitely generated discrete groups, largely motivated by dimension group techniques. The unfaithful extreme harmonic functions (defined only on proper space-time…
Quantum and random walks have been shown to be equivalent in the following sense: a time-dependent random walk can be constructed such that its vertex distribution at all time instants is identical to the vertex distribution of any…
Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there…
One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a…