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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 / 2ᵏ, into an efficient fault-tolerant form using only Hadamard, controlled- not, and / 8 gates. The algorithm runs in O (^2. 71 (1/) ) time, and produces as output a sequence of O (^3. 97 (1/) ) quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within > 0. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU (d).
Dawson et al. (Sun,) studied this question.