Non-Abelian statistics with mixed-boundary punctures on the toric code

The toric code is a simple and exactly solvable example of topological order realizing Abelian anyons. However, it was shown to support nonlocal lattice defects, namely twists, which exhibit non-Abelian anyonic behavior [Phys. Rev. Lett. 105, 030403 (2010)]. Motivated by this result, we investigated the potential of having non-Abelian statistics from puncture defects on the toric code. We demonstrate that an encoding with mixed-boundary punctures reproduces Ising fusion, and a logical Pauli-

X upon their braiding. Our construction paves the way for local lattice defects to exhibit non-Abelian properties that can be employed for quantum information tasks.