In fault-tolerant quantum computation and quantum error-correction one is interested on Pauli matrices that commute with a circuit/unitary. We provide a fast algorithm that decomposes any Clifford gate as a $\textit{minimal}$ product of Clifford transvections. The algorithm can be directly used for finding all Pauli matrices that commute with any given Clifford gate. To achieve this goal, we exploit the structure of the symplectic group with a novel graphical approach.

中文翻译：

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克利福德·盖茨的分解

在容错量子计算和量子纠错中，人们对与电路/ unit元交换的保利矩阵很感兴趣。我们提供了一种快速算法，可将任何Clifford门分解为Clifford对换的$ \ textit {minimal} $乘积。该算法可直接用于查找与任何给定的Clifford门相通的所有Pauli矩阵。为了实现这一目标，我们使用一种新颖的图形方法来开发辛族的结构。