Investigation of continuous-time quantum walk on root lattice $A_n$ and honeycomb lattice

Abstract

The continuous-time quantum walk (CTQW) on root lattice $A_n$ (known as hexagonal lattice for $n=2$) and honeycomb one is investigated by using spectral distribution method. To this aim, some association schemes are constructed from abelian group $Z_m^{\otimes n}$ and two copies of finite hexagonal lattices, such that their underlying graphs tend to root lattice $A_n$ and honeycomb one, as the size of the underlying graphs grows to infinity. The CTQW on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on Terwilliger algebra, where we get the required results for root lattice $A_n$ and honeycomb one, from large enough underlying graphs. Moreover, by using the stationary phase method, the long time behavior of CTQW on infinite graphs is approximated with finite ones. Also it is shown that the Bose-Mesner algebras of our constructed association schemes (called $n$-variable $P$-polynomial) can be generated by $n$ commuting generators, where raising, flat and lowering operators (as elements of Terwilliger algebra) are associated with each generator. A system of $n$-variable orthogonal polynomials which are special cases of \textit{generalized} Gegenbauer polynomials is constructed, where the probability amplitudes are given by integrals over these polynomials or their linear combinations. Finally the suppersymmetric structure of finite honeycomb lattices is revealed. Keywords: underlying graphs of association schemes, continuous-time quantum walk, orthogonal polynomials, spectral distribution. PACs Index: 03.65.Ud

41 pages, 4 figures

References in corpus (3)

• Exponential algorithmic speedup by quantum walk
• Investigation of Continuous-Time Quantum Walk Via Modules of Bose-Mesner and Terwilliger Algebras
• Non-Unitary Quantum Walks on Hyper-Cycles

Cited by in corpus (6)

• Perfect state transfer over distance-regular spin networks
• Recurrences in three-state quantum walks on a plane
• Perfect transference of a d-level quantum state over pseudo-distance-regular networks
• Calculating effective resistances on underlying networks of association schemes
• Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models
• A general algorithm for manipulating non-linear and linear entanglement witnesses by using exact convex optimization