我們利用了泡利群中的元素恰好具有兩個特徵空間，對應於特徵值±1這一事實。我們將+1特徵空間定義為程式碼空間。透過觀察量子態是否位於該程式碼子空間內，我們可以輕鬆驗證是否發生了錯誤。然而，一些運算符可能將量子狀態保留在程式碼空間內，但以不可糾正的方式改變狀態。這些算子是使用群論中的正規化子來定義的，其中克利福德群充當泡利群的正規化子。一個關鍵的結構定理指出，克利福德群與泡利群的商同構於 F2 上的辛群，其元素分解為橫對流的乘積，類似於正交群中的反射。

In error correction, stabilizer formalism is an encoding method that effectively protects against and corrects errors. This method takes advantage of the fact that elements in the Pauli group have exactly two eigenspaces, corresponding to eigenvalues of ±1. We define the +1 eigenspace as the code space. We can easily verify if an error has occurred by observing whether the quantum state resides within this code subspace. However, some operators may keep the quantum state within the code space but alter the state in a way that is not correctable. These operators are defined using the normalizer in group theory, where the Clifford group acts as the normalizer of the Pauli group. A crucial structural theorem states that the quotient of the Clifford group by the Pauli group is isomorphic to the symplectic group over F2, with its elements decomposing into products of transvections, analogs of reflections in the orthogonal group.

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