  Topological properties of quantum systems could provide protection of
information against environmental noise, and thereby drastically advance their
potential in quantum information processing. Most proposals for topologically
protected quantum gates are based on many-body systems, e.g., fractional
quantum Hall states, exotic superconductors, or ensembles of interacting spins,
bearing an inherent conceptual complexity. Here, we propose and study a
topologically protected quantum gate, based on a one-dimensional
single-particle tight-binding model, known as the Su-Schrieffer-Heeger chain.
The proposed $Y$ gate acts in the two-dimensional zero-energy subspace of a Y
junction assembled from three chains, and is based on the spatial exchange of
the defects supporting the zero-energy modes. With numerical simulations, we
demonstrate that the gate is robust against hopping disorder but is corrupted
by disorder in the on-site energy. Then we show that this robustness is
topological protection, and that it arises as a joint consequence of chiral
symmetry, time-reversal symmetry and the spatial separation of the zero-energy
modes bound to the defects. This setup will most likely not lead to a practical
quantum computer, nevertheless it does provide valuable insight to aspects of
topological quantum computing as an elementary minimal model. Since this model
is non-interacting and non-superconducting, its dynamics can be studied
experimentally, e.g., using coupled optical waveguides.
