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 fulfill for superconducting qubits and appears to be a main reason why only single-qubit gates were implemented in a recent experiment [A. A. Abdumalikov Jr. et al., Nature (London) 496, 482 (2013)]. Here we propose to realize nonadiabatic 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 realizing robust holonomic quantum computation with superconducting circuits very promising.

Hybrid quantum systems usually consist of two or more subsystems, which may take the advantages of the different systems. Recently, the hybrid system consisting of circuit electromechanical subsystems have attracted great attention due to its advanced fabrication and scalable integrated photonic circuit techniques. Here, we propose a scheme for high fidelity quantum state transfer between a superconducting qubit and a nitrogen-vacancy center in diamond, which are coupled to a superconducting transmission-line resonator with coupling strength g1 and a nanomechanical resonator with coupling strength g2, respectively. Meanwhile, the two resonators are parametrically coupled with coupling strength J. The system dynamics, including the decoherence effects, is numerical investigated. It is found that both the small () and large () coupling regimes of this hybrid system can not support high fidelity quantum state transfer before significant technique advances. However, in the intermediate coupling regime (J ~ g1 ~ g2), in contrast to a conventional wisdom, high fidelity quantum information transfer can be implemented, providing a promising route towards high fidelity quantum state transfer in similar coupled resonators systems.