  We present a simple method for constructing optimal fault-tolerant
approximations of arbitrary unitary gates using an arbitrary discrete universal
gate set. The method presented is numerical and scales exponentially with the
number of gates used in the approximation. However, for the specific case of
arbitrary single-qubit gates and the fault-tolerant gates permitted by the
concatenated 7-qubit Steane code, we find gate sequences sufficiently long and
accurate to permit the fault-tolerant factoring of numbers thousands of bits
long. A general scaling law of how rapidly these fault-tolerant approximations
converge to arbitrary single-qubit gates is also determined.
