arXiv2025
Symmetry-protected topological phases cannot be described by any local order parameter and are beyond the conventional symmetry-breaking paradigm for understanding quantum matter. They are characterized by topological boundary states robust against perturbations that respect the protecting symmetry. In a clean system without disorder, these edge modes typically only occur for the ground states of systems with a bulk energy gap and would not survive at finite temperatures due to mobile thermal excitations. Here, we report the observation of a distinct type of topological edge modes, which are protected by emergent symmetries and persist even up to infinite temperature, with an array of 100 programmable superconducting qubits. In particular, through digital quantum simulation of the dynamics of a one-dimensional disorder-free "cluster" Hamiltonian, we observe robust long-lived topological edge modes over up to 30 cycles at a wide range of temperatures. By monitoring the propagation of thermal excitations, we show that despite the free mobility of these excitations, their interactions with the edge modes are substantially suppressed in the dimerized regime due to an emergent U(1)$\times$U(1) symmetry, resulting in an unusually prolonged lifetime of the topological edge modes even at infinite temperature. In addition, we exploit these topological edge modes as logical qubits and prepare a logical Bell state, which exhibits persistent coherence in the dimerized and off-resonant regime, despite the system being disorder-free and far from its ground state. Our results establish a viable digital simulation approach to experimentally exploring a variety of finite-temperature topological phases and demonstrate a potential route to construct long-lived robust boundary qubits that survive to infinite temperature in disorder-free systems.