# 【ユニバーサル性】Solovay–Kitaev theoremとGottesman–Knill theorem

## Solovay–Kitaev theorem

https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem

In the mathematical theory of computation, the Solovay–Kitaev theorem says, roughly, that if a set of single-qubitquantum gates generates a densesubset of SU(2) then that set is guaranteed to fill SU(2) quickly, which means good approximations to any desired gate can be created using fairly short sequences of gates from the generating set. It is one of the most important fundamental results in the field of quantum computation.[1]Robert M. Solovay and Alexei Kitaev jointly came up with and proved the theorem.

もし単一量子ビットのゲートセットがSU(2)の稠密集合を生成する場合、そのセットはSU(2)を速やかに満たすことが保証される。つまり、そのようなゲートセットでどんなユニタリゲートもいい近似を作り出せる。

## 稠密集合

https://ja.wikipedia.org/wiki/%E7%A8%A0%E5%AF%86%E9%9B%86%E5%90%88

## SU(2)

n 次の特殊ユニタリ群（とくしゅユニタリぐん、英語: special unitary group）SU(n) とは、行列式が1の n 次ユニタリ行列の為す群の事である。群の演算は行列の積で与えられる。

https://ja.wikipedia.org/wiki/%E7%89%B9%E6%AE%8A%E3%83%A6%E3%83%8B%E3%82%BF%E3%83%AA%E7%BE%A4

## Gottesman–Knill theorem

In quantum computing, the Gottesman–Knill theorem is a theoretical result by Daniel Gottesman and Emanuel Knill that states that stabilizer circuits, circuits that only consist of gates from the normalizer of the qubit Pauli group, also called Clifford group, can be perfectly simulated in polynomial time on a probabilistic classical computer. The Clifford group can be generated solely by using CNOT, Hadamard, and phase gates [1] , therefore stabilizer circuits can be constructed using only these gates.

パウリ演算子もしくはクリフォード群は多項式時間で古典計算機でシミュレートできる。

The reason for the speed up of quantum computers is not yet fully understood. The theorem proves that, for all quantum algorithms with a speed up that relies on entanglement which can be achieved with a CNOT and a Hadamard gate to produce entangled states, this kind of entanglement alone does not give any computing advantage.

CNOTとアダマールゲートから作り出される量子もつれ状態でも、古典計算機より早いとは限らない。

Formal statement

Theorem: A quantum circuit using only the following elements can be simulated efficiently on a classical computer:

Preparation of qubits in computational basis states,

Quantum gates from the Clifford group (Hadamard gates, controlled NOT gates, Phase Gate), and

Measurements in the computational basis.

The Gottesman–Knill theorem shows that even some highly entangled states can be simulated efficiently. Several important types of quantum algorithms use only Clifford gates, most importantly the standard algorithms for entanglement purification and for quantum error correction. From a practical point of view, stabilizer circuits have been simulated in O(n log n) time using the graph state formalism.

## 結論

Non-CliffordのT-ゲートが大事？