Xu Mengke, Li Xi, Wang Xunan, Mi Wanglei, Chen Xiao
School of Information Engineering, China Jiliang University, Hangzhou 310018, China.
School of Software, Henan University, Zhengzhou 450046, China.
Entropy (Basel). 2025 Jun 12;27(6):623. doi: 10.3390/e27060623.
Topological transitions are relevant for boundary conditions. Therefore, we investigate the bulk spectra, edge states, and topological phases of Szegedy's quantum search on a one-dimensional (1D) cycle with self-loops, where the search operator can be formulated as an open boundary condition. By establishing an equivalence with coined quantum walks (QWs), we analytically derive and numerically illustrate the quasienergies dispersion relations of bulk spectra and edge states for Szegedy's quantum search. Interestingly, novel gapless three-band structures are observed, featuring a flat band and three-fold degenerate points. We identify the topological phases ±2 as the Chern number. This invariant is computed by leveraging chiral symmetry in zero diagonal Hermitian Hamiltonians that satisfy our quasienergies constraints. Furthermore, we demonstrate that the edge states enhance searches on the marked vertices, while the nontrivial bulk spectra facilitate ballistic spread for Szegedy's quantum search. Crucially, we find that gapless topological phases arise from three-fold degenerate points and are protected by chiral symmetry, distinguishing ill-defined topological transition boundaries.
拓扑转变与边界条件相关。因此，我们研究了在具有自环的一维（1D）循环上进行塞格迪量子搜索的体谱、边缘态和拓扑相，其中搜索算子可被表述为开放边界条件。通过建立与造币量子行走（QWs）的等价关系，我们解析推导并数值说明了塞格迪量子搜索的体谱和边缘态的准能量色散关系。有趣的是，观察到了新颖的无隙三能带结构，其具有一个平带和三重简并点。我们将拓扑相±2确定为陈数。这个不变量是通过利用满足我们准能量约束的零对角厄米哈密顿量中的手征对称性来计算的。此外，我们证明边缘态增强了对标记顶点的搜索，而非平凡的体谱则促进了塞格迪量子搜索的弹道扩散。至关重要的是，我们发现无隙拓扑相源自三重简并点，并由手征对称性保护，从而区分了不明确的拓扑转变边界。
