  Zero-energy modes localized at the ends of one-dimensional (1D) wires hold
great potential as qubits for fault-tolerant quantum computing. However, all
the candidates known to date exhibit a wave function that decays exponentially
into the bulk and hybridizes with other nearby zero-modes, thus hampering their
use for braiding operations. Here, we show that a quasi-1D diamond-necklace
chain exhibits a completely unforeseen type of robust boundary state, namely
compact localized zero-energy modes that do not decay into the bulk. We
theoretically engineer a lattice geometry to access this mode, and
experimentally realize it in an electronic quantum simulator setup. Our work
provides a general route for the realization of robust and compact localized
zero-energy modes that could potentially be braided without the drawbacks of
hybridization.
