Perfect transmission and parallel composition for quantum walks on graphs with two leads

TL;DR

This paper derives explicit formulas for quantum walk scattering on graphs with leads, characterizes perfect transmission conditions, and introduces additive quantities that simplify the design of graphs with desired transmission properties.

Contribution

It provides a new analytical framework for understanding quantum walk scattering, including explicit formulas and geometric conditions for perfect transmission.

Findings

Explicit formulas for the two-terminal scattering matrix.

Characterization of perfect transmission via additive quantities.

Transmission conditions reduce to a geometric vector-sum problem.

Abstract

We study scattering for continuous-time quantum walks on finite graphs with two attached leads. We derive explicit formulae for the two-terminal scattering matrix in terms of characteristic polynomials of the finite graph and its vertex-deleted subgraphs. For real-weighted two-terminal graphs, we then introduce three real quantities, $\mu_1$, $\mu_2$, and $\nu$, which are each additive under parallel composition of graphs. In these variables, perfect transmission at fixed momentum is characterized by the condition $\mu_1=\mu_2$ together with a hyperbola in the corresponding $(\mu,\nu)$-plane, whose points determine the transmission phase. This turns the search for graphs with prescribed transmission properties into a geometric vector-sum problem for smaller building blocks.