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dbr:Solovay–Kitaev_theorem
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Solovay–Kitaev theorem
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In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that 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, which means any desired gate can be approximated by a fairly short sequence of gates from the generating set. Robert M. Solovay initially announced the result on an email list in 1995, and Alexei Kitaev independently gave an outline of its proof in 1997. Solovay also gave a talk on his result at MSRI in 2000 but it was interrupted by a fire alarm. Christopher M. Dawson and Michael Nielsen call the theorem one of the most important fundamental results in the field of quantum computation.
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In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that 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, which means any desired gate can be approximated by a fairly short sequence of gates from the generating set. Robert M. Solovay initially announced the result on an email list in 1995, and Alexei Kitaev independently gave an outline of its proof in 1997. Solovay also gave a talk on his result at MSRI in 2000 but it was interrupted by a fire alarm. Christopher M. Dawson and Michael Nielsen call the theorem one of the most important fundamental results in the field of quantum computation. 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. 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.
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