The Solovay-Kitaev algorithm
Dawson, C M, Nielsen, M A · CERN Document Server (European Organization for Nuclear Research) · 2005
This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $\\pi / 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\\sc not}, and $\\pi / 8$ gates. The algorithm runs in $O(\\log^{2.71}(1/\\epsilon))$ time, and produces as output a sequence of $O(\\log^{3.97}(1/\\epsilon))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $\\epsilon > 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from $SU(d)$.