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Unformatted text preview: arXiv:quant-ph/0505030 v2 23 Aug 2005 Quantum Information and Computation, Vol. 0, No. 0 (2005) 000–000 c circlecopyrt Rinton Press THE SOLOVAY-KITAEV ALGORITHM CHRISTOPHER M. DAWSON School of Physical Sciences, The University of Queensland, Brisbane, Queensland 4072, Australia MICHAEL A. NIELSEN School of Physical Sciences, The University of Queensland, Brisbane, Queensland 4072, Australia Received May 23, 2006 Revised 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 ). Keywords : Solovay-Kitaev algorithm, universality, fault-tolerance 1 Introduction “I have been impressed by numerous instances of mathematical theories that are really about particular algorithms; these theories are typically formulated in math- ematical terms that are much more cumbersome and less natural than the equiva- lent formulation today’s computer scientists would use.” — Donald E. Knuth  The Solovay-Kitaev (SK) theorem is one of the most important fundamental results in the theory of quantum computation. In its simplest form the SK theorem shows that, roughly speaking, if a set of single-qubit quantum gates generates a dense subset of SU (2), then that set is guaranteed to fill SU (2) quickly , i.e., it is possible to obtain good approximations to any desired gate using surprisingly short sequences of gates from the given generating set. The SK theorem is important if one wishes to apply a wide variety of different single-qubit gates during a quantum algorithm, but is restricted to use gates from a more limited reper- toire. Such a situation arises naturally in the context of fault-tolerant quantum computation, where direct fault-tolerant constructions are typically available only for a limited number of gates (e.g., the Clifford group gates and π/ 8 gate), but one may wish to implement a wider variety of gates, such as the π/ 2 k rotations occurring in Shor’s algorithm [22, 23]. In this sit- uation one must use the limited set of fault-tolerant gates to build up accurate fault-tolerant 1 2 The Solovay-Kitaev algorithm .. ....
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