The Solovay-Kitaev algorithm

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 π/2 k , into an efficient fault-tolerant form using only Hadamard, controlled-not, and π/8 gates. The algorithm runs in O(log 2.71 (1/ǫ)) time, and produces as output a sequence of O(log 3.97 (1/ǫ)) quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within ǫ > 0. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU (d).