Generalized eigenfunctions for quantum walks via path counting approach

We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

A Path-Counter Method for Fault-Tolerant Minimal Routing Algorithms in 2D Mesh

Fault-tolerant Manhattan routing algorithms aim at finding a Manhattan path between the source and destination nodes and route around all faulty nodes. However, besides faulty nodes, some nonfaulty nodes that are helpless to make up a fault-tolerant Manhattan path should also be routed around. How to label such nonfaulty nodes efficiently is a major challenge. We propose a path-counter method. It can label such nodes with low time-complexity by counting every node’s fault-tolerant Manhattan paths to the source or destination node. During the path-counting procedure, no available nodes will be sacrificed under arbitrary fault distribution. Compared with fault-block model based work, our proposed method is independent of fault distribution, so its computational complexity is very low.