Moments of coinless quantum walks on lattices
- 146 Downloads
The properties of the coinless quantum-walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps, but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than random-walk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on d-dimensional lattices by employing the methods of Fourier transforms and generating functions. The mean square displacement for large times is explicitly calculated for the one- and two-dimensional lattices, and using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.
KeywordsCoinless quantum walks Moments Mean square displacement Standard deviation
RAMS acknowledges financial support from Capes-Faperj E-45/2013. RP thanks Faperj (Grant No. E-26/102.350/2013) and CNPq (Grant Nos. 304709/2011-5, 474143/2013-9, and 400216/2014-0). SB acknowledges financial support from the US National Science Foundation through Grant DMR-1207431.
- 4.Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing. STOC ’01, pp. 37–49. ACM, New York, NY, USA (2001)Google Scholar
- 5.Nayak, A., Vishwanath, A.: Quantum Walk on the Line (2000). arXiv:quant-ph/0010117v1
- 14.Falk, M.: Quantum search on the spatial grid (2013). arXiv:quant-ph/1303.4127
- 16.Ambainis, A., Portugal, R., Nahimov, N.: Spatial search on grids with minimum memory. arXiv:quant-ph/1312.0172
- 17.Portugal, R., Boettcher, S., Falkner, S.: One-dimensional coinless quantum walks (2014). arXiv:1408.5166v2 [quant-ph]