The
Solovay-Kitaev algorithm
(pp081-095)
Christopher M. Dawson and Michael A.
Nielsen
doi:
https://doi.org/10.26421/QIC6.1-6
Abstracts:
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).
Key words:
Solovay-Kitaev algorithm, universality,
fault-tolerance |