Incoherent tunneling effects in a one-dimensional quantum walk

Abstract

In this article we investigate the effects of shifting position decoherence, arisen from the tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discreet time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not fade, in contrary to the coin decoherence for which the walker undergos the quantum-toclassical transition even for weak noise. Particularly, we show that the quadratic dependency of the variance on the time and also the coin-position entanglement, i.e. two important quantum aspects of the coherent quantum walk, are preserved in the presence of tunneling decoherence. Furthermore, we present an explicit expression for the probability distribution of decoherent one-dimensional quantum walk in terms of the corresponding coherent probabilities, and show that this type of decoherence smooths the probability distribution.

Möbius quantum walk

“…If a walker, walks along the length of this strip, it can visit both sides of the strip without crossing the edges. In fact, Möbius strip is a surface (defined by a normal vector to the surface) with only one side and only one boundary [21]. In our model, |r plays the role of the normal vector to the surface of strip, so any node on the circumference of a circle not only has a specific index for its location but also a normal vector for the surface which it is located on.…”

“…The quadratic behavior of QW variance is a direct consequence of the unitary evolution of coherent QWs, but, in practice, the isolation of a QW system from its environment is impossible. It has been shown that environmental effects and noises make the evolution non-unitary and decoherency occurs [4][5][6], So the variance of a QW (t 2 ) may transit to the classical one (t) [7,8].…”

Analytical expression for variance of homogeneous-position quantum walk with decoherent position

“…Although weak noise can enhance the properties of quantum walk in such a way that to be useful and desirable for developing quantum algorithms [27], it is shown that increasing the noise level, eliminates all quantum properties and changes it to a classical system [26]. One may concludes that any kind of noise can eliminate the quantum properties and change the system to a classical one, but it is shown that if the environmental noise allows the walker to tunnel out into its closest neighbor with a defined probability, not only the quantum walk does not * annabestani@shahroodut.ac.ir transit to a classical one, but also it helps to improve quantum properties [28].…”