Here we study the performance of a concatenated error-detection code in a system that permits only nearest-neighbor interactions in one dimension. We make use of a new message-passing scheme which maximizes the number of errors that can be reliably corrected by the code. Our numerical results indicate that arbitrarily accurate universal quantum computation is possible if the probability of failure of each elementary physical operation is below approximately 10^{-5}. This threshold is three orders of magnitude lower than the highest known.
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