Alternate two-dimensional quantum walk with a single-qubit coin

Abstract

We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the non-localized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states, in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of x-y spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.

Extraordinary behaviors in a two-dimensional decoherent alternative quantum walk

“…It has been verified that when the walkers start from the position |0 x |0 y , with an appropriate choice of the initial coin state, the position distribution of the 2D AQW at time t is as same as that from the 2D DTQW with the four-level Grover coin [35,36].…”

“…In this alternative quantum walk (AQW), the two-level coin affects the walker moving in the x-direction at first, followed by the motion of walker along the y-direction. The position distribution pattern induced by the 2D DTQW with four-level Grover coin can be recovered with the AQW with only one two-level coin [35,36]. Due to the function of the coin as the register of the coherence and randomness in the aforementioned DTQW, when the searching space increases to n-dimension, we need a 2 n -level coin to implement the search process.…”

“…Recently, a 2D quantum walk with one two-level coin is presented [35][36][37][38][39]. In this alternative quantum walk (AQW), the two-level coin affects the walker moving in the x-direction at first, followed by the motion of walker along the y-direction.…”

Implementation of a spatial two-dimensional quantum random walk with tunable decoherence

“…The shapes obtained in Fig. 3 correspond to the probability distributions of a two-dimensional Grover walk [19,25]. The walker propagates with ballistic speed, characterized by a quadratic dependence of the variance V (n) with the step index, that is, V (n) ≈ n 2 .…”

“…[20]. Another approach is to mimic the two-dimensional QRW evolution by performing two subsequent one-dimensional QRWs [24,25].…”