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.
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 . This can be done such that both and are close to the identity (since ). 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 with an exhaustive search of bounded-length gate sequences.
Bouland, Adam; Giurgica-Tiron, Tudor (2021-12-03), Efficient Universal Quantum Compilation: An Inverse-free Solovay-Kitaev Algorithm, arXiv:2112.02040 Kuperberg, Greg (2023-06-22), "Breaking the cubic barrier in the Solovay-Kitaev algorithm", arXiv:2306.13158 [quant-ph]