  Quantum computation that combines the coherence stabilization virtues of
decoherence-free subspaces and the fault tolerance of geometric holonomic
control is of great practical importance. Some schemes of adiabatic holonomic
quantum computation in decoherence-free subspaces have been proposed in the
past few years. However, non-adiabatic holonomic quantum computation in
decoherence-free subspaces, which avoids long run-time requirement but with all
the robust advantages, remains an open problem. Here, we demonstrate how to
realize non-adiabatic holonomic quantum computation in decoherence-free
subspaces. By using only three neighboring physical qubits undergoing
collective dephasing to encode one logical qubit, we realize a universal set of
quantum gates.
