  Geometric phases are noise-resilient, and thus provide a robust way towards
high fidelity quantum manipulation. Here we experimentally demonstrate
arbitrary non-adiabatic holonomic single-qubit quantum gates for both a
superconducting transmon qubit and a microwave cavity in a single-loop way. In
both cases, an auxiliary state is utilized, and two resonant microwave drives
are simultaneously applied with well-controlled but varying amplitudes and
phases for the arbitrariness of the gate. The resulting gates on the transmon
qubit achieve a fidelity of 0.996 characterized by randomized benchmarking and
the ones on the cavity show an averaged fidelity of 0.978 based on a full
quantum process tomography. In principle, a nontrivial two-qubit holonomic gate
between the qubit and the cavity can also be realized based on our presented
experimental scheme. Our experiment thus paves the way towards practical
non-adiabatic holonomic quantum manipulation with both qubits and cavities in a
superconducting circuit.
