Anomoulas Non-Abelian Statistics for Non-Hermitian Majorana Zero Modes: Alternative Approach to Universal Topological Quantum Computation

In condensed matter physics, non-Abelian statistics for Majorana zero modes (or Majorana Fermions) is very important, really exotic, and completely robust. The race for searching Majorana zero modes and verifying the corresponding non-Abelian statistics becomes an important frontier in condensed matter physics. In this letter, we generalize the Majorana zero modes to non-Hermitian (NH) topological systems that show universal but quite different properties from their Hermitian counterparts. Based on the NH Majorana zero modes, the orthogonal and nonlocal Majorana qubits are well defined. In particular, the non-Abelian statistics for these NH Majorana zero modes become anomalous, which is different from the usual non-Abelian statistics. The usual Ivanovs braiding operator for two Majorana modes is generalized to a non-Hermitian Ivanovs braiding perator. The one-dimensional NH Kitaev model is taken as an example to numerically verify the anomalous non-Abelian statistics for two NH Majorana zero modes. The numerical results are exactly consistent with the theoretical prediction. With the help of braiding these two zero modes, the $pi/8$ gate can be reached and thus universal topological quantum computation becomes possible.