Quantum Random Walks in One Dimension

Norio Konno, Takuya Machida

Recently Mc Gettrick introduced and studied a discrete-time 2-state quantum walk (QW) with a memory in one dimension. He gave an expression for the amplitude of the QW by path counting method. Moreover he showed that the return probability of the walk is more than 1/2 for any even time. In this paper, we compute the stationary distribution by considering the walk as a 4-state QW without memory. Our result is consistent with his claim. In addition, we obtain the weak limit theorem of the rescaled QW. This behavior is strikingly different from the corresponding classical random walk and the usual 2-state QW without memory as his numerical simulations suggested.

Yuliy M. Baryshnikov, Wil Brady, Andrew Bressler +1 more

Norio Konno

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2 times 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.

Norio Konno

Peng Xue

Quantum random walks have been much studied recently,largely due to their highly non-classical behaviors.In this paper,we review the classical and quantum random walks of one-dimensional,and analyze two classical two-dimensional models and compare them.Furthermore we propose a two-dimensional quantum random walk algorithm.We compute the standard deviation,thereby proving our two-dimensional algorithm is superior to that of one-dimension.We propose a scheme,which is based on single photon with pairs of qubits,to implement it.

Luigi Accardi, G. S. Watson