Realization of a holonomic quantum computer in a chain of three-level systems

•We develop a novel scheme for universal holonomic quantum computation.•The scheme involves non-Abelian geometric phases in a spin-chain.•The resources scale linearly with the number of logical qubits.•The scheme does not require adiabatic evolution.

Holonomic quantum computation is the idea to use non-Abelian geometric phases to implement universal quantum gates that are robust to fluctuations in control parameters. Here, we propose a compact design for a holonomic quantum computer based on coupled three-level systems. The scheme does not require adiabatic evolution and can be implemented in arrays of atoms or ions trapped in tailored standing wave potentials.