A | B | C | D | E | F | G | H | CH | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
In quantum information and computation, the Solovay–Kitaev theorem says that if a set of single-qubit quantum gates generates a dense subgroup of SU(2), then that set can be used to approximate any desired quantum gate with a short sequence of gates that can also be found efficiently. This theorem is considered one of the most significant results in the field of quantum computation and was first announced by Robert M. Solovay in 1995 and independently proven by Alexei Kitaev in 1997.[1][2] Michael Nielsen and Christopher M. Dawson have noted its importance in the field.[3]
A consequence of this theorem is that a quantum circuit of constant-qubit gates can be approximated to error (in operator norm) by a quantum circuit of gates from a desired finite universal gate set.[4] By comparison, just knowing that a gate set is universal only implies that constant-qubit gates can be approximated by a finite circuit from the gate set, with no bound on its length. So, the Solovay–Kitaev theorem shows that this approximation can be made surprisingly efficient, thereby justifying that quantum computers need only implement a finite number of gates to gain the full power of quantum computation.
Statement
Let be a finite set of elements in SU(2) containing its own inverses (so implies ) and such that the group they generate is dense in SU(2). Consider some . Then there is a constant such that for any , there is a sequence of gates from of length such that . That is, approximates to operator norm error.[3] Furthermore, there is an efficient algorithm to find such a sequence. More generally, the theorem also holds in SU(d) for any fixed d.
This theorem also holds without the assumption that contains its own inverses, although presently with a larger value of that also increases with the dimension .[5]
Quantitative bounds
The constant can be made to be for any fixed .[6] However, there exist particular gate sets for which we can take , which makes the length of the gate sequence optimal up to a constant factor.[7]
Proof idea
Every known proof of the fully general Solovay–Kitaev theorem proceeds by recursively constructing a gate sequence giving increasingly good approximations to .[3] Suppose we have an approximation such that . Our goal is to find a sequence of gates approximating to error, for . By concatenating this sequence of gates with , we get a sequence of gates such that .
The main idea in the original argument of Solovay and Kitaev is that commutators of elements close to the identity can be approximated "better-than-expected". Specifically, for satisfying and and approximations satisfying and , then
where the big O notation hides higher-order terms. One can naively bound the above expression to be , but the group commutator structure creates substantial error cancellation.
We can use this observation to approximate as a group commutator
Antropológia
Aplikované vedy
Bibliometria
Dejiny vedy
Encyklopédie
Filozofia vedy
Forenzné vedy
Humanitné vedy
Knižničná veda
Kryogenika
Kryptológia
Kulturológia
Literárna veda
Medzidisciplinárne oblasti
Metódy kvantitatívnej analýzy
Metavedy
Metodika
Text je dostupný za podmienok Creative
Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších
podmienok.
Podrobnejšie informácie nájdete na stránke Podmienky
použitia.
www.astronomia.sk | www.biologia.sk | www.botanika.sk | www.dejiny.sk | www.economy.sk | www.elektrotechnika.sk | www.estetika.sk | www.farmakologia.sk | www.filozofia.sk | Fyzika | www.futurologia.sk | www.genetika.sk | www.chemia.sk | www.lingvistika.sk | www.politologia.sk | www.psychologia.sk | www.sexuologia.sk | www.sociologia.sk | www.veda.sk I www.zoologia.sk