Persistent Quantum Memory in Iterated Lifts

Abstract

We study quantum coherence in continuous-time quantum walks on perfect graphs generated by the symmetric lift ${\mathrm{HL}}'_2(G)$, a canonical, unweighted, undirected construction defined as the line graph of a bipartite double cover of $G$. This lift acts as both a coherence-preserving and coherence-inducing transformation: it preserves and scales structured quantum interference in highly symmetric base graphs, and induces sustained coherence in random or weakly structured ones. In small graphs such as $K_4$, $K_5$, and the Petersen graph, where quantum walks exhibit sharp revivals and high return probability, repeated $\mathrm{HL}'_2$ lifting produces towers of perfect graphs with thousands to tens of thousands of vertices that retain periodic or quasi-periodic coherence. When applied to random regular or Erd\H{o}s--R\'enyi graphs with flat or decaying return behavior, the lift introduces structured interference and significant amplification of mean and peak return probabilities. To quantify these effects, we evaluate standard coherence metrics from quantum resource theory, including inverse participation ratio (IPR), purity, relative entropy of coherence, and the logarithmic coherence number. These measures confirm that $\mathrm{HL}'_2$ lifting delocalizes eigenstates, increases coherence entropy, and expands the basis support of quantum states. These results demonstrate that $\mathrm{HL}'_2$ is a scalable and structurally grounded mechanism for organizing quantum interference, and introduce a new family of perfect graphs that support long-time quantum coherence without spectral tuning or engineered weights.