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Communications in Theoretical Physics

Note on Implementation of Three-Qubit SWAP Gate

Hai‐Rui Wei, Yao-Min Di, Yan Wang, Jie Zhang

Published Jan 1, 2010• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/53/1/18

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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.
We investigate the realization of 2-qutrit logic gate in a bipartite 3-level system with qusi-Ising interaction. On the basis of Cartan decomposition of matrices,the unitary matrices of 2-qutrit are factorized into products of a series of realizable matrices.It is equivalent to exerting a certain control field on the system,and the control goal is usually gained by a sequence of control pulses.The general discussion on the realization of 2-qutrit logic gate is made first,and then the realization of the ternary SWAP gate and the ternry SWAP~(1/2)gate are discussed specifically,and the sequences of control pulses and drift processes implementing these gates are given.
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。
We investigate the realization of 2-qutrit logic gate in a bipartite 3-level system with qusi-Ising interaction. On the basis of Cartan decomposition of matrices, the unitary matrices of 2-qutrit are factorized into products of a series of realizable matrices. It is equivalent to exerting a certain control field on the system, and the control goal is usually gained by a sequence of control pulses. The general discussion on the realization of 2-qutrit logic gate is made first, and then the realization of the ternary SWAP gate and the ternary gate are discussed specifically, and the sequences of control pulses and drift processes implementing these gates are given.
In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.
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.
We consider a model of two interacting always-on, exchange-only qubits for which controlled phase (CPHASE), controlled NOT (CNOT), quantum Fourier transform (QFT) and SWAP operations can be implemented only in a few electrical pulses in a nanosecond time scale. Each qubit is built of three quantum dots (TQD) in a triangular geometry with three electron spins which are always kept coupled by exchange interactions only. The qubit states are encoded in a doublet subspace and are fully electrically controlled by a voltage applied to gate electrodes. The two qubit quantum gates are realized by short electrical pulses which change the triangular symmetry of TQD and switch on exchange interaction between the qubits. We found an optimal configuration to implement the CPHASE gate by a single pulse of the order 2.3 ns. Using this gate, in combination with single qubit operations, we searched for optimal conditions to perform the other gates: CNOT, QFT and SWAP. Our studies take into account environment effects and leakage processes as well. The results suggest that the system can be implemented for fault tolerant quantum computations.