Continuous-Time Quantum State Transfer with a Generalized Laplacian
Yujia Shi
TL;DR
The paper investigates continuous-time quantum walks driven by the generalized Laplacian $L_k = A + kD$ to overcome fidelity limitations on general graphs. By leveraging cospectrality and graph symmetries (involution), it shows that tuning $k$ concentrates dynamics onto a two-dimensional subspace spanned by two endpoint-localized eigenvectors, enabling high-fidelity transfer between designated vertices. It provides explicit relationships between fidelity and $k$ (via $Q = k(d_1 - d_2)$) and proves bounds guaranteeing $F(Q) > 1-\epsilon$ with polynomially bounded readout times; examples include $K_{2,n-2}$ and the path $P_n$. This work extends vertex-weighted and signless Laplacian analyses to the generalized Laplacian, offering a practical mechanism for robust quantum routing on graphs. The results have potential implications for designing quantum communication protocols over networked systems where graph structure and vertex degrees play a pivotal role.
Abstract
Quantum walks generated by the adjacency matrix or the Laplacian are known to exhibit low transfer fidelity on general graphs. In this paper, we study continuous-time quantum walks governed by the generalized Laplacian operator L_k = A+kD, where A is the adjacency matrix, D is the degree matrix, and k is a real-valued parameter. Recent work of Duda, McLaughlin, and Wong showed that in the single-excitation Heisenberg (XYZ) spin model, one can realize walks generated by this family of operators on signed weighted graphs. Motivated by earlier studies on vertex-weighted graphs, we demonstrate that for certain graphs, tuning the parameter k can significantly enhance the fidelity of state transfer between endpoints.