  Von Neumann's classic "multiplexing" method is unique in achieving
high-threshold fault-tolerant classical computation (FTCC), but has several
significant barriers to implementation: i) the extremely complex circuits
required by randomized connections, ii) the difficulty of calculating its
performance in practical regimes of both code size and logical error rate, and
iii) the (perceived) need for large code sizes. Here we present numerical
results indicating that the third assertion is false, and introduce a novel
scheme that eliminates the two remaining problems while retaining a threshold
very close to von Neumann's ideal of 1/6. We present a simple, highly ordered
wiring structure that vastly reduces the circuit complexity, demonstrates that
randomization is unnecessary, and provides a feasible method to calculate the
performance. This in turn allows us to show that the scheme requires only
moderate code sizes, vastly outperforms concatenation schemes, and under a
standard error model a unitary implementation realizes universal FTCC with an
accuracy threshold of p < 5.5%, in which p is the error probability for 3-qubit
gates. FTCC is a key component in realizing measurement-free protocols for
quantum information processing. In view of this we use our scheme to show that
all-unitary quantum circuits can reproduce any measurement-based feedback
process in which the asymptotic error probabilities for the measurement and
feedback are 0.51p and 1.51p, respectively.
