Modifying quantum walks: A scattering theory approach
arXiv:0705.4612 · doi:10.1088/1751-8113/40/37/011
Abstract
We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering theory. The probability of entering a graph from one tail and leaving from another can be found from the scattering matrix of the graph. We show how the scattering matrix of a graph that is an automorphic image of the original is related to the scattering matrix of the original graph, and we show how the scattering matrix of the reverse graph is related to that of the original graph. Modifications of graphs and the effects of these modifications are then considered. In particular we show how the scattering matrix of a graph is changed if we remove two tails and replace them with an edge or cut an edge and add two tails. This allows us to combine graphs, that is if we connect two graphs we can construct the scattering matrix of the combined graph from those of its parts. Finally, using these techniques, we show how two graphs can be compared by constructing a larger larger graph in which the two original graphs are in parallel, and performing a quantum walk on the larger graph. This is a kind of quantum walk interferometry.
22 pages and 1 figure
References in corpus (4)
Cited by in corpus (27)
- Quantum walks: a comprehensive review
- Recurrence properties of unbiased coined quantum walks on infinite -dimensional lattices
- Scattering theory of topological phases in discrete-time quantum walks
- Perfect state transfer by means of discrete-time quantum walk on complete bipartite graphs
- Group transformations and entangled-state quantum gates with directionally unbiased linear-optical multiports
- Equivalence between discrete quantum walk models in arbitrary topologies
- Dynamical system induced by quantum walk
- Quantum simulation of discrete-time Hamiltonians using directionally unbiased linear optical multiports
- Enhanced-sensitivity interferometry with phase-sensitive unbiased multiports
- Joint Entanglement of Topology and Polarization Enables Error-Protected Quantum Registers
- Quantum scattering theory on graphs with tails
- Quantum simulation of topologically protected states using directionally unbiased linear-optical multiports
- Electric circuit induced by quantum walk
- Directionally-Unbiased Unitary Optical Devices in Discrete-Time Quantum Walks
- Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models
- Quantum walk based state transfer algorithms on the complete M-partite graph
- Green function approach for scattering quantum walks
- Generalized eigenfunctions for quantum walks via path counting approach
- A Simple Method for Finding the Scattering Coefficients of Quantum Graphs
- Superdiffusivity of quantum walks: A Feynman sum-over-paths description
- Explicit expression of scattering operator of some quantum walks on impurities
- Asymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks
- Design for implementation of discrete-time quantum walk with circulant matrix on graph by optical polarizing elements
- Analytical solutions for quantum walks on 1D chain with different shift operators
- A comfortable graph structure for Grover walk
- A discontinuity of the energy of quantum walk in impurities
- Relation between quantum walks with tails and quantum walks with sinks on finite graphs