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. {\bf 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.

9 pages, 9 figures, RevTeX4

References in corpus (7)

• Universal computation by quantum walk
• Quantum Walk in Position Space with Single Optically Trapped Atoms
• Discrete single-photon quantum walks with tunable decoherence
• Faster quantum walk algorithm for the two dimensional spatial search
• Limit distributions of two-dimensional quantum walks
• Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin
• Recurrence properties of unbiased coined quantum walks on infinite $d$-dimensional lattices

Cited by in corpus (33)

• Quantum walks: a comprehensive review
• Two-dimensional topological quantum walks in the momentum space of structured light
• Experimental realisation of a delayed-choice quantum walk
• The Dirac equation as a quantum walk: higher dimensions, observational convergence
• Two quantum walkers sharing coins
• Quantum walks and quantum search on graphene lattices
• Implementation of a spatial two-dimensional quantum random walk with tunable decoherence
• Electric quantum walks in two dimensions
• Extraordinary behaviors in two-dimensional decoherent alternative quantum walk
• Generic Quantum Walks with Memory
• Localization and limit laws of a three-state alternate quantum walk on a two-dimensional lattice
• N-dimensional alternate coined quantum walks from a dispersion relation perspective
• Decomposition of split-step quantum walks for simulating Majorana modes and edge states
• Weak limit theorem for a one-dimensional split-step quantum walk
• One dimensional lazy quantum walks and occupancy rate
• Quantum walk on a cylinder
• Strongly trapped two-dimensional quantum walks
• Decoherence on a two-dimensional quantum walk using four- and two-state particle
• Localization-like effect in two-dimensional alternate quantum walks with periodic coin operations
• Understanding and controlling N-dimensional quantum walks via dispersion relations. Application to the 2D and 3D Grover walks: Diabolical points and more
• Sensitivity of quantum walks to boundary of two-dimensional lattices: approaches from the CGMV method and topological phases
• A quantum walk with a delocalized initial state: contribution from a coin-flip operator
• Limit theorems and localization of three state quantum walks on a line defined by generalized Grover coins
• Controlled Alternate Quantum Walks based Quantum Hash Function
• Discrete-Time Quantum Walks on Oriented Graphs
• A limit law of the return probability for a quantum walk on a hexagonal lattice
• Designing three-way entangled and nonlocal two-way entangled single particle states via alternate quantum walks
• Entanglement transfer, accumulation and retrieval via quantum-walk-based qubit-qudit dynamics
• Quantum walk on a square lattice with identical particles
• A quantum walk on the half line with a particular initial state
• Two-dimensional quantum central limit theorem by quantum walks
• Quantum Coin Flipping, Qubit Measurement and Generalized Fibonacci Numbers
• Stationary amplitudes of quantum walks on the higher-dimensional integer lattice