Fig. 4: Quantum processor measurements of HOT corner modes on the staggered cubic lattice.

a The header row displays energy spectra for the topologically trivial C0 and inequivalent nontrivial C4a, C4b, and C8 configurations. The configurations host 0, 4, and 8 midgap zero modes (red diamonds), as measured via IQPE on an 18-qubit chain plus an ancillary qubit; the shaded red band indicates the IQPE energy resolution. Schematics illustrating the locations of topologically robust corners are shown on the right. Subsequent rows depict the time-evolution of five initial states on a 6 × 6 × 6 lattice mapped onto an 18-qubit chain—localized at a corner, on an edge, on a face, and in the bulk of the cube, and delocalized in the vicinity of a corner. The leftmost column plots 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, z) of the initial state (darker shading represents higher density). The central grid shows occupancy density measured on hardware at a later time (t = 0.6), for the corresponding initial state (row) and lattice configuration (column). Error bars represent standard deviation across repetitions on different qubit chains and devices. Again, initial states localized close to topological corners exhibit higher occupational fidelity. b Hamiltonian schematic of the interacting chain realizing a minimal 4 × 4 × 4 cubic lattice. Sites on the chain are colored black; colored vertices connecting to multiple sites on the chain denote interaction terms. Intra- and inter-cell hoppings, mapped onto interactions, are respectively denoted \({v}_{{{{{{{{\boldsymbol{\pi }}}}}}}}}^{\alpha }\) and \({w}_{{{{{{{{\boldsymbol{\pi }}}}}}}}}^{\alpha }\) for axes α \({w}_{{{{{{{{\boldsymbol{\pi }}}}}}}}}^{\alpha }\) {x, y, z} and parities \({{{{{{{\boldsymbol{\pi }}}}}}}}\in {{\mathbb{Z}}}_{2}^{2}\).