  Holonomies, arising from non-Abelian geometric transformations of quantum
states in Hilbert space, offer a promising way for quantum computation. These
holonomies are not commutable and thus can be used for the realization of a
universal set of quantum logic gates, where the global geometric feature may
result in some noise-resilient advantages. Here we report the first on-chip
realization of a non-Abelian geometric controlled-Not gate in a superconducting
circuit, which is a building block for constructing a holonomic quantum
computer. The conditional dynamics is achieved in an all-to-all connected
architecture involving multiple frequency-tunable superconducting qubits
controllably coupled to a resonator; a holonomic gate between any two qubits
can be implemented by tuning their frequencies on resonance with the resonator
and applying a two-tone drive to one of them. This gate represents an important
step towards the all-geometric realization of scalable quantum computation on a
superconducting platform.
