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Chinese Physics Letters

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

Hai‐Rui Wei, Yao-Min Di, ZHANG-Jie

Published Sep 1, 2008• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/25/9/004

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

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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]

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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.
We prove that a generic three-qubit quantum logic gate can be implemented using at most 98 one-qubit rotations about the yyy- and zzz-axes and 40 CNOT gates, beating an earlier bound of 64 CNOT gates.
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.
Quantum circuits currently constitute a dominant model for quantum computation according to M. Nielsen and I. Chuang (2000). Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and a small number of elementary quantum gates by A. Barenco et al. (1995) and G. Song and A. Klappenecker (2003) as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worst-case upper bound to 23 elementary gates, of which at most 4 (CNOTs) entail multi-qubit interactions. Our analysis shows that previous known synthesis algorithms, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm according to G. Cybenko (2001) has a worst case of 61 gates of which 18 may be CNOTs. Our technique rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis. They are related to the canonical decomposition of two-qubit gate with respect to the "magic basis" of phase-shifted Bell states by N. Khaneja et al. (2001) and M. Lewenstein et al. (2001). We extend this decomposition in terms of elementary gates for quantum computation.
This paper proposes an optimized quantum block-ZXZZXZZXZ decomposition method that results in more optimal quantum circuits than the quantum Shannon decomposition, which was presented in 2005 by M. M\"ott\"onen, and J. J. Vartiainen [in Trends in quantum computing research, edited by S. Shannon (Nova Science Publishers, 2006) Chap. 7, p. 149, arXiv:quant-ph/0504100]. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only single-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general three-qubit gate can be decomposed using 19 cnot gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}{4}^{n}\ensuremath{-}\frac{3}{2}{2}^{n}+\frac{5}{3}$ cnot gates, which is less than the best-known exact decomposition algorithm by $({4}^{n\ensuremath{-}2}\ensuremath{-}1)/3$ cnot gates.
This paper proposes a new optimized quantum block-ZXZ decomposition method [7,8,10] that results in more optimal quantum circuits than the quantum Shannon decomposition (QSD)[27], which was introduced in 2006 by Shende et al. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only one-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general 3-qubit gate can be decomposed using 19 CNOT gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have 22484n−322n+53\frac{22}{48}4^n - \frac{3}{2}2^n +\frac{5}{3}4822​4n−23​2n+35​ CNOT gates, which is less that the best known exact decomposition algorithm by (4n−2−1)/3(4^{n-2} -1)/3(4n−2−1)/3 CNOT gates.