标题： 置换群的量子电路实现 显示英文标题

标题： Realization of permutation groups by quantum circuit

摘要： 本文中，我们专门利用CNOT门来实现由多于两个元素生成的置换群。在引理1中，我们回顾到执行一个两量子比特交换门操作既需要也只需三个CNOT门。随后，在引理2中，我们表明执行n量子比特替换操作所需的CNOT门的最大数量为3(n-1)。继续前进，我们在第3节的分析揭示了使用五个或更少的CNOT门不足以实现与置换元素(123)对应的三量子比特交换门。因此，对于(123)，六个CNOT门既必要又足够。这是通过采用图论方法严格验证最多五个CNOT门的结果来实现的。使用计算工具，我们彻底探索了所有包含恰好六个CNOT门的有效电路图，以成功执行(123)的交换门，通过解释引理6和表2中的等价类来实现。我们在定理7中得出结论。为了将我们的分析扩展到多量子比特场景，我们在定义8中给出了可约和不可约置换元素。我们在定理9中澄清了多量子比特空间中行之间的等价性，并提供了多量子比特进行上述操作的近似上界。本文的全面探索旨在为进一步通过多次使用特定的两量子比特门来理解量子电路优化铺平道路。 显示英文摘要

摘要： In this paper, we exclusively utilize CNOT gates for implementing permutation groups generated by more than two elements. In Lemma 1, we recall that three CNOT gates are both necessary and sufficient to execute a two-qubit swap gate operation. Subsequently, in Lemma 2, we show that the maximum number of CNOT gates needed to carry out an n-qubit substitution operation is 3(n-1). Moving forward, our analysis in Section 3 reveals that utilizing five or fewer CNOT gates is insufficient for implementing a three-qubit swap gate corresponding to the permutation element (123). Hence six CNOT gates are both necessary and sufficient for implementing (123). This is done by employing a graph-theoretic approach to rigorously validate the results in terms of at most five CNOT gates. Using computational tools, we exhaustively explore all valid circuit diagrams containing exactly six CNOT gates to successfully execute the swap gate for (123), by explaining the equivalence classes in Remark 6 and Table 2. We conclude them in Theorem 7.To extend our analysis to the multiqubit scenario, we present the reducible and irreducible permutation elements in Definition 8. We clarify the equivalence between rows in the multi-qubit space and provide an approximate upper bound for multi-qubits to perform the aforementioned operations in Theorem 9. The comprehensive exploration of this paper aims to pave the way for further advancements in understanding quantum circuit optimization via multiple use of a specific two-qubit gate.