Quantum walks (QWs) provide a powerful tool as a quantum simulator to study and understand topological phases. Using such a quantum simulator, some topological phenomena have been discussed. However, all the experimental observations on the topological phenomena in QWs have been restricted to evolution in one dimension (1D) so far. The existing 2D experimental platforms cannot be applied to study topological phenomena due to lack of full control in the position space. Thus, some interesting topological phenomena in the 2D QW that do not exist in the 1D case, e.g., the edge-state-enhanced transport, have not been demonstrated experimentally. Here we report the experimental realization of 2D QW using spatial positions and orbital angular momentum of light. Based on our constructed experimental platform, we have observed 2D topological bound states with vanishing Chern numbers and confirmed the robustness of these bound states with respect to perturbations and disorder, which go beyond what has been known in static systems and are unique to periodically driven systems. Our studies not only represent an important advance in the study of topological phases, but also open up an avenue to explore topological properties in multidimensional QWs.
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 resource-intensive split-step or split-coin quantum walk protocols. Odd and even-site cyclic graphs exhibit markedly different spectral characteristics, with rotationally symmetric flat bands emerging exclusively in
4n-site graphs (
n∈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 as well as remain stable against phase-preserving perturbations and are independent of initial states. This scheme serves as a resource-efficient and versatile platform to engineer topological phases, transitions, edge states, and flat bands in small-scale quantum systems, opening new avenues for robust quantum memory, protected state transfer, and compact implementations of fault-tolerant quantum technologies.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.