dorsal/arxiv
View SchemaThe Solovay-Kitaev algorithm
| Authors | Christopher M. Dawson, Michael A. Nielsen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0505030 |
| URL | https://arxiv.org/abs/quant-ph/0505030 |
Abstract
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 $SU(d)$.
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"abstract": "This pedagogical review presents the proof of the Solovay-Kitaev theorem in\nthe form of an efficient classical algorithm for compiling an arbitrary\nsingle-qubit gate into a sequence of gates from a fixed and finite set. The\nalgorithm can be used, for example, to compile Shor\u0027s algorithm, which uses\nrotations of $\\pi / 2^k$, into an efficient fault-tolerant form using only\nHadamard, controlled-{\\sc not}, and $\\pi / 8$ gates. The algorithm runs in\n$O(\\log^{2.71}(1/\\epsilon))$ time, and produces as output a sequence of\n$O(\\log^{3.97}(1/\\epsilon))$ quantum gates which is guaranteed to approximate\nthe desired quantum gate to an accuracy within $\\epsilon \u003e 0$. We also explain\nhow the algorithm can be generalized to apply to multi-qubit gates and to gates\nfrom $SU(d)$.",
"arxiv_id": "quant-ph/0505030",
"authors": [
"Christopher M. Dawson",
"Michael A. Nielsen"
],
"categories": [
"quant-ph"
],
"title": "The Solovay-Kitaev algorithm",
"url": "https://arxiv.org/abs/quant-ph/0505030"
},
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"source": {
"execution_id": "c98d2a52-4b15-4973-8810-81d050ae0870",
"id": "arXiv Dataset IDs",
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"variant": "snapshot-2026-03-01",
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