Hierarchical Majoranas in a Programmable Nanowire Network

We propose a hierarchical architecture for building "logical" Majorana zero modes using "physical" Majorana zero modes at the Y-junctions of a hexagonal network of semiconductor nanowires. Each Y-junction contains three "physical" Majoranas, which hybridize when placed in close proximity, yielding a single effective Majorana mode near zero energy. The hybridization of effective Majorana modes on neighboring Y-junctions is controlled by applied gate voltages on the links of the honeycomb network. This gives rise to a tunable tight-binding model of effective Majorana modes. We show that selecting the gate voltages that generate a Kekul\'e vortex pattern in the set of hybridization amplitudes yields an emergent "logical" Majorana zero mode bound to the vortex core. The position of a logical Majorana can be tuned adiabatically, \textit{without} moving any of the "physical" Majoranas or closing any energy gaps, by programming the values of the gate voltages to change as functions of time. A nanowire network supporting multiple such "logical" Majorana zero modes provides a physical platform for performing adiabatic non-Abelian braiding operations in a fully controllable manner.