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Chinese Physics Letters

Modified Khaneja-Glaser Decomposition and Realization of Three-Qubit Quantum Gate

Hai‐Rui Wei, 狄尧民, 张洁

Published Jan 1, 2008• 0 citations
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Abstract

量门的最佳的实现为量计算的实现是关键的。我们稍微为 n-qubits 修改 Khaneja-Glaser 分解(KGD ) 并且在分解的第二步给新 Cartan subalgbra。基于这修改 KGD，我们调查 three-qubit 逻辑门的实现并且获得一扇一般 three-qubit 量逻辑门能被实现至多关于 y 和 z 斧子和 26 扇 CNOT 门使用 73 one-qubit 门旋转的结果。[从作者抽象]

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量门的最佳的实现为量计算的实现是关键的。我们稍微为 n-qubits 修改 Khaneja-Glaser 分解(KGD ) 并且在分解的第二步给新 Cartan subalgbra。基于这修改 KGD，我们调查 three-qubit 逻辑门的实现并且获得一扇一般 three-qubit 量逻辑门能被实现至多关于 y 和 z 斧子和 26 扇 CNOT 门使用 73 one-qubit 门旋转的结果。[从作者抽象]
我们与 bequsi 相互作用在一个由两部组成的 3 水平系统调查 2-qutrit 逻辑门的实现。根据矩阵的 Cartan 分解， 2-qutrit 的单一的矩阵是进一系列可实现的矩阵的产品的 factorized。它等价于在系统，和控制目标上施加某个控制域被控制脉搏的一个序列通常获得。逻辑门被成为第一的 2-qutrit 的实现的一般讨论，然后实现第三交换门，第三的门明确地被讨论，并且控制脉搏的序列和实现这些门的飘移过程被给。
Optimal implementation of quantum gates is crucial for realization of quantum computation. We slightly modify the Khaneja-Glaser decomposition (KGD) for n-qubits and give a new Cartan subalgbra in the second step of the decomposition. Based on this modified KGD, we investigate the realization of three-qubit logic gate and obtain the result that a general three-qubit quantum logic gate can be implemented using at most 73 one-qubit gates rotations with respect to the y and z axes and 26 CNOT gates.
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。
Recently, Vatan and Williams utilize a matrix decomposition of SU(2n)SU(2^n)SU(2n) introduced by Khaneja and Glaser to produce {\tt CNOT}-efficient circuits for arbitrary three-qubit unitary evolutions. In this note, we place the Khaneja Glaser Decomposition ({\tt KGD}) in context as a SU(2n)=KAKSU(2^n)=KAKSU(2n)=KAK decomposition by proving that its Cartan involution is type {\bf AIII}, given n≥3n \geq 3n≥3. The standard type {\bf AIII} involution produces the Cosine-Sine Decomposition (CSD), a well-known decomposition in numerical linear algebra which may be computed using mature, stable algorithms. In the course of our proof that the new decomposition is type {\bf AIII}, we further establish the following. Khaneja and Glaser allow for a particular degree of freedom, namely the choice of a commutative algebra a\mathfrak{a}a, in their construction. Let χ1n\chi_1^nχ1n​ be a {\tt SWAP} gate applied on qubits 111, nnn. Then χ1nvχ1n=k1  a  k2\chi_1^n v \chi_1^n=k_1\; a \; k_2χ1n​vχ1n​=k1​ak2​ is a KGD for $\mathfrak{a}=\mbox{span}_{\mathbb{R}} \{ \chi_1^n ( \ket{j}\bra{N-j-1} -\ket{N-j-1}\bra{j}) \chi_1^n \}$ if and only if v=(χ1nk1χ1n)(χ1naχ1n)(χ1nk2χ1n)v=(\chi_1^n k_1 \chi_1^n) (\chi_1^n a \chi_1^n)(\chi_1^n k_2 \chi_1^n)v=(χ1n​k1​χ1n​)(χ1n​aχ1n​)(χ1n​k2​χ1n​) is a CSD.