标题： 量子行走揭示循环图中的拓扑平带、鲁棒边缘态和拓扑相变 显示英文标题

标题： Quantum walks reveal topological flat bands, robust edge states and topological phase transitions in cyclic graphs

摘要： 拓扑相、边缘态和平坦带在合成量子系统中是拓扑量子计算和抗噪声信息处理的关键资源。我们提出了一种基于循环图上步依赖量子行走的方案，称为循环量子行走（CQWs），利用离散傅里叶变换和有效哈密顿量来模拟奇异的拓扑现象。我们的方法能够生成有能隙和无能隙的拓扑相，包括类似狄拉克锥的能量色散、拓扑非平凡的平坦带和受保护的边缘态，而无需采用分步或分币协议。奇数位和偶数位循环图表现出显著不同的光谱特性，其中旋转对称的平坦带仅在$4n$-位图（$n\in \mathbf{N}$）中出现。我们分析性地建立了拓扑、有能隙平坦带出现的条件，并表明旋转空间中的能隙闭合意味着动量空间中狄拉克锥的形成。此外，我们在奇数和偶数循环图之间不同拓扑相的界面处设计了受保护的边缘态。我们数值证明边缘态对中等静态和动态门干扰具有鲁棒性，并且对保持相位的扰动保持稳定。该方案为在量子系统中工程拓扑相、相变、边缘态和平坦带提供了一个资源高效且多功能的平台，为容错量子技术开辟了新途径。 显示英文摘要

摘要： Topological phases, edge states, and flat bands in synthetic quantum systems are a key resource for topological quantum computing and noise-resilient information processing. We introduce a scheme based on step-dependent quantum walks on cyclic graphs, termed cyclic quantum walks (CQWs), to simulate exotic topological phenomena using discrete Fourier transforms and an effective Hamiltonian. Our approach enables the generation of both gapped and gapless topological phases, including Dirac cone-like energy dispersions, topologically nontrivial flat bands, and protected edge states, all without resorting to split-step or split-coin protocols. Odd and even-site cyclic graphs exhibit markedly different spectral characteristics, with rotationally symmetric flat bands emerging exclusively in $4n$-site graphs ($n\in \mathbf{N}$). We analytically establish the conditions for the emergence of topological, gapped flat bands and show that gap closings in rotation space imply the formation of Dirac cones in momentum space. Further, we engineer protected edge states at the interface between distinct topological phases in both odd and even cycle graphs. We numerically demonstrate that the edge states are robust against moderate static and dynamic gate disorder and remain stable against phase-preserving perturbations. This scheme serves as a resource-efficient and versatile platform for engineering topological phases, transitions, edge states, and flat bands in quantum systems, opening new avenues for fault-tolerant quantum technologies.