Fig. 3: Quantum processor measurements of 2D HOT zero modes and their roles in preserving state fidelity.

a Ordered eigenenergies on a 10 × 10 lattice for the topologically trivial C0 and nontrivial C2 and C4 configurations. They correspond to 0, 2, and 4 midgap zero modes (red diamonds), as measured via IQPE on a 20-qubit quantum chain plus an additional ancillary qubit; the shaded red band indicates the IQPE energy resolution. The corner state profiles (right insets) and other eigenenergies (black and gray dots) are numerically obtained via ED. Time-evolution of four initial states on a 16 × 16 lattice mapped onto a 32-qubit chain—b, c localized at corners to highlight topological distinction, d localized along an edge, and e delocalized in the vicinity of a corner. Left plots show occupancy fidelity for the various lattice configurations, obtained from ED and quantum hardware (labeled HW), with insets showing the site-resolved occupancy density ρ(x, y) of the initial states (darker shading represents higher density). The right grid shows occupancy density measured on hardware at two later times. States with good overlap with robust corners exhibit minimal evolution. Error bars represent standard deviation across repetitions on different qubit chains and devices. In general, the heavy overlap between an initial state and a HOT eigenstate confers topological robustness, resulting in significantly slowed decay. f Schematic of the interacting chain Hamiltonian, mapped from the parent 2D lattice, illustrated for a smaller 6 × 6 square lattice. The physical sites of the interacting boson chain are colored black, with their many-body interactions represented by colored vertices. Intra- and inter-cell hoppings, mapped onto interactions, are respectively denoted \({v}_{{{{{{{{\boldsymbol{\pi }}}}}}}}}^{\alpha }\) and \({w}_{{{{{{{{\boldsymbol{\pi }}}}}}}}}^{\alpha }\) for axes α \(\in\) {x, y} and parities \({{{{{{{\boldsymbol{\pi }}}}}}}}\in {{\mathbb{Z}}}_{2}^{1}\).