Non-Abelian statistics for bosonic symmetry-protected topological phases

Symmetry-protected non-Abelian (SPNA) statistics opens new frontiers in quantum statistics and enriches the schemes for topological quantum computing. In this work, we propose a new paradigm of SPNA statistics in one-dimensional correlated bosonic symmetry-protected topological (SPT) phases and uncover exotic universal features from a systematic investigation. In particular, we show that for generic bosonic SPT phases described by real Hamiltonians, the SPNA statistics of topological zero modes fall into two distinct classes. The first class exhibits conventional braiding of hard-core bosonic zero modes. Furthermore, we discover a second class of unconventional braiding statistics characterized by a nonlinear transformation, featuring a fractionalization of the first class and reminiscent of the non-Abelian statistics of symmetry-protected Majorana pairs. The two distinct classes of statistics have topological origin in classifying non-Abelian Berry phases for braiding processes of real-Hamiltonian systems, distinguished by whether the holonomy involves a reflection operation. To illustrate, we focus on a specific bosonic SPT phase with particle-hole symmetry, and demonstrate that both classes of braiding statistics can be feasibly realized in tri-junctions with and without the aid of a controlled defect, respectively. Analytic and numerical results are given. We demonstrate how to encode logical qubits and implement both single- and two-qubit gates using the two classes of SPNA statistics. Finally, we propose feasible experimental schemes to observe these predictions and identify the parameter regimes for the high-fidelity braiding, paving the way for the experimental observation of our results in the near future.