Charge-$4e$ superconductor with parafermionic vortices: A path to universal topological quantum computation

Abstract

Topological superconductors (TSCs) provide a promising route to fault-tolerant quantum information processing. However, the canonical Majorana platform based on $2e$ TSCs remains computationally constrained. In this work, we find a $4e$ TSC that overcomes these constraints by combining a charge-$4e$ condensate with an Abelian chiral $\mathbb{Z}_3$ topological order in an intertwined fashion. Remarkably, this $4e$ TSC can be obtained by proliferating vortex-antivortex pairs in a stack of two $2e$ $p+ip$ TSCs, or by melting a $ν=2/3$ quantum Hall state. Specific to this TSC, the $hc/(4e)$ fluxes act as charge-conjugation defects in the topological order, whose braiding with anyons transmutes anyons into their antiparticles. This symmetry enrichment leads to $\mathbb{Z}_3$ parafermion zero modes trapped in the elementary vortex cores, which naturally encode qutrits. Braiding the parafermion defects alone generates the full many-qutrit Clifford group. We further show that a single-probe interferometric measurement enables topologically protected magic-state preparation, promoting Clifford operations to a universal gate set. Because the non-Abelian modes are bound to flux defects, they can, in principle, be externally controlled using superconducting circuit-based technology. More broadly, our results highlight hierarchical electron aggregation, the formation and condensation of higher-charge electron clusters, as a design principle for topological quantum matter with increased computational capability.