  Nonadiabatic geometric quantum computation in decoherence-free subspaces has
received increasing attention due to the merits of its high-speed
implementation and robustness against both control errors and decoherence.
However, all the previous schemes in this direction have been based on the
conventional geometric phases, of which the dynamical phases need to be
removed. In this paper, we put forward a scheme of nonadiabatic geometric
quantum computation in decoherence-free subspaces based on unconventional
geometric phases, of which the dynamical phases do not need to be removed.
Specifically, by using three physical qubits undergoing collective dephasing to
encode one logical qubit, we realize a universal set of geometric gates
nonadiabatically and unconventionally. Our scheme not only maintains all the
merits of nonadiabatic geometric quantum computation in decoherence-free
subspaces, but also avoids the additional operations required in the
conventional schemes to cancel the dynamical phases.
