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arXiv · 2605.14640

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

Allan John Gerrard·Ryo Asaka·Kazumitsu Sakai
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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.

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Allan John Gerrard, Ryo Asaka, Kazumitsu Sakai. 2026-05-14. Perfect transmission and parallel composition for quantum walks on graphs with two leads. https://arxiv.org/abs/2605.14640

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