  We introduce and implement a technique to extend the quantum computational
power of cluster states by replacing some projective measurements with
generalized quantum measurements (POVMs). As an experimental demonstration we
fully realize an arbitrary three-qubit cluster computation by implementing a
tunable linear-optical POVM, as well as fast active feedforward, on a two-qubit
photonic cluster state. Over 206 diferent computations, the average output
fidelity is 0.9832 +/- 0.0002; furthermore the error contribution from our POVM
device and feedforward is only of order 10e-3, less than some recent thresholds
for fault-tolerant cluster computing.
