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