  We show that thresholds for fault-tolerant quantum computation are solely
determined by the quality of single-system operations if one allows for
d-dimensional systems with $8 \leq d \leq 32$. Each system serves to store one
logical qubit and additional auxiliary dimensions are used to create and purify
entanglement between systems. Physical, possibly probabilistic two-system
operations with error rates up to 2/3 are still tolerable to realize
deterministic high quality two-qubit gates on the logical qubits. The
achievable error rate is of the same order of magnitude as of the single-system
operations. We investigate possible implementations of our scheme for several
physical set-ups.
