Ultrametric Quantum Computing: Tree-Topology Error Correction

Abstract

The observation that positional notation carries a native ultrametric tree structure extends into quantum computing through the tensor-product architecture of qubit registers. We formalize three theses: (I) a tree-topology quantum processor confines errors to subtrees via the strong triangle inequality, elevating the error-correction threshold; (II) a 3D-integrated superconducting architecture realizing a 7-ary tree of depth 3 (343 physical qubits) is manufacturable with current fabrication technology; (III) an on-chip bank of prime-frequency resonators enables spectral engineering of the tree's error-confinement properties. We implement a numerical simulation comparing tree-topology and grid-topology codes, deriving threshold estimates and error-propagation statistics. The chapter extends the Ultrametric Foundation thesis (§12) with formal derivations, architecture specifications, simulation results, and a mapping to the broader research program. ---