Asymptotic behavior of quantum walks on the line
arXiv:1108.1878 · doi:10.1016/j.jfa.2011.12.016
Abstract
This paper gives various asymptotic formulae for the transition probability associated with discrete time quantum walks on the real line. The formulae depend heavily on the `normalized' position of the walk. When the position is in the support of the weak-limit distribution obtained by Konno, one observes, in addition to the limit distribution itself, an oscillating phenomenon in the leading term of the asymptotic formula. When the position lies outside of the support, one can establish an asymptotic formula of large deviation type. The rate function, which expresses the exponential decay rate, is explicitly given. Around the boundary of the support of the limit distribution (called the `wall'), the asymptotic formula is described in terms of the Airy function.
32 pages, 1 figure
Cited by in corpus (21)
- Localization of the Grover walks on spidernets and free Meixner laws
- Weak limit theorem for a one-dimensional split-step quantum walk
- Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator
- Feynman checkers: towards algorithmic quantum theory
- Bandit Algorithm Driven by a Classical Random Walk and a Quantum Walk
- Eigenvalues, absolute continuity and localizations for periodic unitary transition operators
- Exponential tail estimates for quantum lattice dynamics
- Weak limit theorem of a two-phase quantum walk with one defect
- Feynman checkers: number-theoretic properties
- Asymptotic stability of small bound state of nonlinear quantum walks
- Resolvent Methods for Quantum Walks with an Application to a Thue-Morse Quantum Walk
- Directivity of quantum walk via its random walk replica
- Feynman checkers: external electromagnetic field and asymptotic properties
- An eigenfunction expansion formula for one-dimensional two-state quantum walks
- Scattering and inverse scattering for nonlinear quantum walks
- Time operators for continuous-time and discrete-time quantum walks
- The Hamiltonians generating one-dimensional discrete-time quantum walks
- An algebraic structure for one-dimensional quantum walks and a new proof of the weak limit theorem
- A crossover between open quantum random walks to quantum walks
- Eigenbasis of the Evolution Operator of 2-Tessellable Quantum Walks
- Skeleton structure inherent in discrete-time quantum walks