To implement a set of universal quantum logic gates based on non-Abelian
geometric phases, it is a conventional wisdom that quantum systems beyond two
levels are required, which is extremely difficult to fulfil for superconducting
qubits, appearing to be a main reason why only single qubit gates was
implemented in a recent experiment [A. A. Abdumalikov Jr \emph{et al}., Nature
496, 482 (2013)]. Here we propose to realize non-adiabatic holonomic quantum
computation in decoherence-free subspace on circuit QED, where one can use only
the two levels in transmon qubits, a usual interaction, and a minimal resource
for the decoherence-free subspace encoding. In particular, our scheme not only
overcomes the difficulties encountered in previous studies, but also can still
achieve considerably large effective coupling strength, such that high fidelity
quantum gates can be achieved. Therefore, the present scheme makes it very
promising way to realize robust holonomic quantum computation with
superconducting circuits.Comment: V4: published version; V1: submitted on April
