We analyze and study the effects of locality on the fault-tolerance threshold
for quantum computation. We analytically estimate how the threshold will depend
on a scale parameter r which estimates the scale-up in the size of the circuit
due to encoding. We carry out a detailed semi-numerical threshold analysis for
concatenated coding using the 7-qubit CSS code in the local and `nonlocal'
setting. First, we find that the threshold in the local model for the [[7,1,3]]
code has a 1/r dependence, which is in correspondence with our analytical
estimate. Second, the threshold, beyond the 1/r dependence, does not depend too
strongly on the noise levels for transporting qubits. Beyond these results, we
find that it is important to look at more than one level of concatenation in
order to estimate the threshold and that it may be beneficial in certain
places, like in the transportation of qubits, to do error correction only
infrequently.Comment: REVTeX, 44 pages, 19 figures, to appear in Physical Review