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理论物理通讯：英文版

Note on Implementation of Three-Qubit SWAP Gate

Hai‐Rui Wei, 狄尧民, WANG Yan, 张沽

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

在这份报纸， three-quhit 的合成和实现交换门被讨论。three-quhit 交换门能被分解成 2 two-qubit 的产品交换门，并且它能由 6 扇 CNOT 门被认识到。尽管结果很简单，研究说明了那，为 multi-qubit 门的矩阵分解的当前的方法不能得到那。然后 three-qubit 的实现与 Ising 在三个旋转系统交换门相互作用被调查，控制脉搏和飘移过程的顺序被给实现门。它需要 23 控制脉搏和 12 个飘移过程。因为相互作用不能随意断断续续地被交换， three-qubit 的实现也在特定的量系统交换门不能简单地下来到 2 two-qubit 交换门。

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理论物理通讯：英文版•Jan 1, 2016•0 citations
在这份报纸， three-quhit 的合成和实现交换门被讨论。three-quhit 交换门能被分解成 2 two-qubit 的产品交换门，并且它能由 6 扇 CNOT 门被认识到。尽管结果很简单，研究说明了那，为 multi-qubit 门的矩阵分解的当前的方法不能得到那。然后 three-qubit 的实现与 Ising 在三个旋转系统交换门相互作用被调查，控制脉搏和飘移过程的顺序被给实现门。它需要 23 控制脉搏和 12 个飘移过程。因为相互作用不能随意断断续续地被交换， three-qubit 的实现也在特定的量系统交换门不能简单地下来到 2 two-qubit 交换门。
我们调查海森堡旋转簇，二个粒子由一个时间依赖者磁场控制了。系统被调节三步的时间依赖者磁场的振幅，频率，和相互作用时间控制。然后，我们解决二粒子的系统的时间依赖者 Schrodinger 方程，并且获得时间进化操作符。由 three-timestep 相互作用，波浪功能从起始的状态演变到最后的状态，和全部的进化操作员能被表示为三个进化操作员的一个产品。由调整物理参数，关键 two-qubit 逻辑门， C 不门，能身体上被认识到。
In this paper, the synthesis and implementation of three-quhit SWAP gate is discussed. The three-quhit SWAP gate can be decomposed into product of 2 two-qubit SWAP gates, and it can be realized by 6 CNOT gates. Research illustrated that although the result is very simple, the current methods of matrix decomposition for multi-qubit gate can not get that. Then the implementation of three-qubit SWAP gate in the three spin system with Ising interaction is investigated and the sequence of control pulse and drift process to implement the gate is given. It needs 23 control pulses and 12 drift processes. Since the interaction can not be switched on and off at will, the realization of three-qubit SWAP gate in specific quantum system also can not simply come down to 2 two-qubit SWAP gates.
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.