量子计算机中的身份根之间的转换
Translating between the roots of the identity in quantum computers
摘要：
Clifford+ $T$量子计算门库可以创建由组$\langle H, T\rangle$生成的所有酉矩阵。矩阵$T$可以被认为是Pauli $Z$的第四根，因为$T^4 = Z$或者也是恒等式$I$的第八根。阿达玛矩阵$H$可用于在泡利矩阵之间进行转换，因为$(HTH)^4$给出泡利矩阵$X$。我们推广泡利矩阵的这些根（或单位根）和平移矩阵，以研究它们生成的群：所谓的泡利根群。在这项工作中，我们介绍了这样的群体的形式化，研究有限性和无限性的性质，并精确地确定平等和子群关系。
英文摘要：
The Clifford+$T$ quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group $\langle H, T\rangle$. The matrix $T$ can be considered the fourth root of Pauli $Z$, since $T^4 = Z$ or also the eighth root of the identity $I$. The Hadamard matrix $H$ can be used to translate between the Pauli matrices, since $(HTH)^4$ gives Pauli $X$. We are generalizing both these roots of the Pauli matrices (or roots of the identity) and translation matrices to investigate the groups they generate: the so-called Pauli root groups. In this work we introduce a formalization of such groups, study finiteness and infiniteness properties, and precisely determine equality and subgroup relations.
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