  In many physical approaches to quantum computation, error-correction schemes
assume the ability to form two-dimensional qubit arrays with nearest-neighbor
couplings and parallel operations at multiple qubit sites. While semiconductor
spin qubits exhibit long coherence times relative to their operation speed and
single-qubit fidelities above error correction thresholds, multiqubit
operations in two-dimensional arrays have been limited by fabrication,
operation, and readout challenges. We present a two-by-two array of four
singlet-triplet qubits in gallium arsenide and show simultaneous coherent
operations and four-qubit measurements via exchange oscillations and
frequency-multiplexed single-shot measurements. A larger multielectron quantum
dot is fabricated in the center of the array as a tunable interqubit link,
which we utilize to demonstrate coherent spin exchange with selected qubits.
Our techniques are extensible to other materials, indicating a path towards
quantum processors with gate-controlled spin qubits.
