Structural Graph-Theoretic Methods for Quantum State Transfer in Finite and Bounded Infinite Networks

Mohammadesmail Nikfar¹

1) Ph.D. Graduate in Mathematics, Doctor of Philosophy (Ph.D.) in Pure Mathematics, Mathematics Teacher, District 4 Department of Education, Tehran, Iran

IOI Code : ACCD-ACAABB
Publication : 10th International Conference on Applied Researches in Science & Engineering (carseconf.com)
Abstract :
    The study of quantum-state transfer on weighted graphs constitutes a fundamental intersection of spectral graph theory, operator theory, and quantum information dynamics. In the continuous-time quantum walk framework, a countable weighted graph with uniformly bounded absolute vertex-weight sums induces a bounded adjacency operator, thereby providing a mathematically well-defined Hamiltonian for the associated unitary evolution. This work develops a unified structural and spectral approach to the construction and analysis of quantum networks exhibiting prescribed state-transfer phenomena, with particular emphasis on equitable partitions, twin-vertex configurations, and localized modifications of edge weights. A general structural criterion is established to characterize circumstances under which suitable equitable decompositions, combined with twin-vertex arrangements, generate nontrivial quantum state transfer in bounded weighted graphs. The resulting methodology accommodates both finite graphs and countably infinite networks under appropriate operator-boundedness conditions. Among its constructive consequences, the framework yields infinitely many connected, finite, unweighted graphs of maximum degree k, for every integer k≥3, that exhibit pair-state transfer at prescribed evolution times. Corresponding constructions are developed for signed graphs containing exactly one negatively weighted edge, while their underlying unsigned graphs retain the prescribed maximum-degree constraint. These signed configurations support plus-state transfer at the designated times. Further constructions establish perfect state transfer between a plus state and a pair state, broadening the range of realizable transfer mechanisms. The analysis also identifies extensive classes of networks supporting sedentary pair states and sedentary plus states, thereby elucidating additional forms of constrained quantum evolution. Beyond explicit graph constructions, probabilistic and asymptotic techniques are employed to investigate the prevalence of the relevant transfer properties within families of planar graphs and trees. Under the specified asymptotic framework, the proportion of connected finite unweighted planar graphs possessing the considered pair-state-transfer property converges to one as the number of vertices tends to infinity. An analogous asymptotic result is established for plus-state transfer in networks obtained through a localized negative-edge modification. This sign alteration can also facilitate perfect state transfer between a plus state and a pair state. Related asymptotic conclusions are derived for finite unweighted trees, extending the investigation of transfer phenomena to another fundamental class of graph structures. To further expand the range of realizable network architectures, several systematic graph-generation techniques are examined, including graph blow-ups, Cayley-type constructions, and the attachment of path-like tails. These operations produce additional families of finite and bounded infinite unweighted graphs, together with signed variants, that exhibit prescribed pair-state transfer, plus-state transfer, perfect state transfer, and associated dynamical behaviors. Collectively, the results establish a comprehensive constructive framework in which equitable graph partitions, twin-vertex symmetries, localized edge-sign modifications, spectral analysis, asymptotic graph enumeration, and graph transformations operate as complementary mechanisms for engineering quantum-state transfer. This framework advances the structural understanding of continuous-time quantum dynamics on discrete networks and provides a unified mathematical perspective on state-transfer phenomena in both finite and suitably bounded countably infinite graph systems.

Keywords : continuous time quantum walks quantum state transfer spectral graph theory graph hamiltonians bounded adjacency operators equitable partitions twin vertex structures weighted graphs signed graphs pair state transfer plus state transfer perfect state transfer sedentary quantum states countably infinite networks planar graph asymptotics tree networks cayley graphs graph blow ups graph transformations quantum information dynamics