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 then that set is guaranteed to fill SU 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. 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 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.
Statement
Let be a finite set of elements in SU containing its own inverses. 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.
The proof of the Solovay-Kitaev theorem proceeds by recursively constructing a gate sequence giving increasingly good approximations to. 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 key idea 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 use this observation by rewriting the expression we wish to approximate as a group commutator. This can be done such that both and are close to the identity. So, if we recursively compute gate sequences approximating and to error, we get a gate sequence approximating to the desired better precision with . We can get a base case approximation with constant by brute-force computation of all sufficiently long gate sequences.
