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## NTT Communication Science Laboratories, Nippon Telegraph and Telephone Corporation (2005)

### Citations

2492 |
Quantum computation and quantum information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ... The other involves approximate strategies by which an arbitrary unitary matrix is decomposed approximately into a sequence of a fixed set of elementary gates, as shown in Solovay-Kitaev theorem (cf. =-=[1]-=-, Appendix 3). In this paper, we treat exact decomposition. Cosine-Sine decomposition (CSD), which is a well-known algorithm in numerical linear algebra, was the first algorithm utilized for this purp... |

1792 |
Differential Geometry, Lie Groups and Symmetric Spaces
- Helgason
- 1978
(Show Context)
Citation Context ...al. [4] and by Shende et al [6]. On the other hand, Khaneja and Glaser provided another kind of decomposition [8], which was later named KGD. KGD lies within the framework of the G = KAK theorem (cf. =-=[9]-=-, Theorem 8.6) in Lie group theory. This theorem shows that an element g ∈ SU(2 n ) is decomposed into matrix products k1ak2 for some k1, k2 ∈ exp(k) and a ∈ exp(h). Here, su(2 n ) = k ⊕ m is a Cartan... |

65 |
et al., “Elementary Gates for Quantum Computation
- Barenco
- 1995
(Show Context)
Citation Context ...are available. Some CSD-based algorithms [3–6] have been investigated with the aim of improving Barenco’s result that an arbitrary 2 n × 2 n unitary matrix is composed of O(n 2 4 n ) elementary gates =-=[7]-=-. And improvement to O(4 n ) elementary gates has been reported by Möttönen et al. [4] and by Shende et al [6]. On the other hand, Khaneja and Glaser provided another kind of decomposition [8], which ... |

18 |
Time optimal control in spin systems, Phys
- Khaneja, Brockett, et al.
- 2001
(Show Context)
Citation Context ...of bases of k, m, and h in Ref. [8] so that the selection matches an NMR system, and they proved that a time-optimal control on a two-qubit NMR quantum computer can be obtained from the decomposition =-=[10]-=-. Thus, KGD can be regarded as the G = KAK theorem on the particular bases. It should be noted that KGD does not give a unique translation of the input matrix into a quantum circuit. Bullock [11] show... |

15 |
A Rudimentary Quantum Compiler
- Tucci
(Show Context)
Citation Context ...pendix 3). In this paper, we treat exact decomposition. Cosine-Sine decomposition (CSD), which is a well-known algorithm in numerical linear algebra, was the first algorithm utilized for this purpose =-=[2]-=-. CSD applies the well-known algorithm for computing generalized singular value decomposition (GSVD). In CSD, we first divide an input matrix g into four square matrices and then apply SVD to each mat... |

13 |
Knapp: Lie Groups: Beyond an introduction
- W
- 1996
(Show Context)
Citation Context ... m is a Cartan decomposition in Lie algebra su(2 n ), k and m = k ⊥ are orthogonal vector spaces contained in su(2 n ), and h is a maximal Abelian subalgebra (a Cartan subalgebra) contained in m (cf. =-=[15]-=-, §VI.2). Matrices k1, a, and k2 are not uniquely determined from g. They depend on the selections of the bases of k, m, and h; besides, they are not determined even if bases are selected. Khaneja and... |

11 | Markov: “Arbitrary Two-Qubit Computation in 23
- Bullock, L
- 2003
(Show Context)
Citation Context ...rtan involutions and square root matrices. Here, we select Θ(X) as σ1zXσ1z for CSD and as σnzXσnz for KGD, where X ∈ SU(2 n ). The strategy utilized in our algorithm is related to those used in Refs. =-=[12,13]-=-. However, those strategies provided methods for computing typeAII KAK decomposition; no translation between type-AII decompositions and type-AIII decompositions was provided. Furthermore, the method ... |

9 |
Quantum Circuits for General Multiqubit
- Vartiainen, Möttönen, et al.
- 2004
(Show Context)
Citation Context ...improving Barenco’s result that an arbitrary 2 n × 2 n unitary matrix is composed of O(n 2 4 n ) elementary gates [7]. And improvement to O(4 n ) elementary gates has been reported by Möttönen et al. =-=[4]-=- and by Shende et al [6]. On the other hand, Khaneja and Glaser provided another kind of decomposition [8], which was later named KGD. KGD lies within the framework of the G = KAK theorem (cf. [9], Th... |

9 | Cartan Decomposition of SU(2 n ), Constructive Controllability of Spin Systems and Universal Quantum Computing,” LANL ePrint quant-ph/0010100
- Khaneja, Glasser
- 2001
(Show Context)
Citation Context ...y gates [7]. And improvement to O(4 n ) elementary gates has been reported by Möttönen et al. [4] and by Shende et al [6]. On the other hand, Khaneja and Glaser provided another kind of decomposition =-=[8]-=-, which was later named KGD. KGD lies within the framework of the G = KAK theorem (cf. [9], Theorem 8.6) in Lie group theory. This theorem shows that an element g ∈ SU(2 n ) is decomposed into matrix ... |

8 | Note on the Khaneja Glaser decomposition
- Bullock
(Show Context)
Citation Context ...tion [10]. Thus, KGD can be regarded as the G = KAK theorem on the particular bases. It should be noted that KGD does not give a unique translation of the input matrix into a quantum circuit. Bullock =-=[11]-=- showed that CSD can also be regarded in the framework of the G = KAK theorem; i.e., CSD uses the type-AIII KAK decomposition with the global Cartan decomposition Θ defined as Θ(X) = σ1zXσ1z for X ∈ S... |

7 | Decompositions of General Quantum Gates
- Möttönen, Vartiainen
- 2006
(Show Context)
Citation Context ...t, ∈ SU(4) (ℓ, j = 1 or 2) are applied selectively; that is, g (0) 1 and g (0) 2 are applied when the third qubit is |0〉, the symbol of the control qubit represents the uniformly controlled rotations =-=[5,16]-=-. g (j−1) ℓ whereas g (1) 1 and g (1) 2 are applied when it is |1〉. Fig. 2 shows the image of a decomposition in (2) for a three-qubit system; that is, we choose Θ as in (6) and a Cartan subalgebra as... |

7 |
Quantum fast Fourier transform viewed as a special case of recursive application of cosine-sine decomposition
- Tucci
(Show Context)
Citation Context ...s, it is difficult to formulate a class of matrices u1, u2, v1, and v2 such that relation (1) holds for a given class of input matrices. Actually, to reproduce the well-known QFT circuit by using CSD =-=[2, 14]-=-, Tucci changes the rows and columns of the QFT matrices beforehand and makes each submatrix hold a convenient form, which can be written by the (n − 1)-qubit QFT. It would not be possible to describe... |

7 |
Transformation of quantum states using uniformly controlled rotations
- Vartiainen, Möttönen, et al.
- 2005
(Show Context)
Citation Context ...t, ∈ SU(4) (ℓ, j = 1 or 2) are applied selectively; that is, g (0) 1 and g (0) 2 are applied when the third qubit is |0〉, the symbol of the control qubit represents the uniformly controlled rotations =-=[5,16]-=-. g (j−1) ℓ whereas g (1) 1 and g (1) 2 are applied when it is |1〉. Fig. 2 shows the image of a decomposition in (2) for a three-qubit system; that is, we choose Θ as in (6) and a Cartan subalgebra as... |

4 | Time reversal and n-qubit Canonical Decompositions. xxx.lanl.gov quant-ph/0402051
- Bullock, Brennen, et al.
(Show Context)
Citation Context ...rtan involutions and square root matrices. Here, we select Θ(X) as σ1zXσ1z for CSD and as σnzXσnz for KGD, where X ∈ SU(2 n ). The strategy utilized in our algorithm is related to those used in Refs. =-=[12,13]-=-. However, those strategies provided methods for computing typeAII KAK decomposition; no translation between type-AII decompositions and type-AIII decompositions was provided. Furthermore, the method ... |

2 |
Compiling quantum circuits into elementary unitary operations
- Svore
- 2004
(Show Context)
Citation Context ...e algebra su(2 n ). And let the global Cartan involution (cf. [15], p. 362) of SU(2n ) be Θ. Then k and m have the following property: { { x if x ∈ k X if X ∈ exp(k) θ(x) = , Θ(X) = −x if x ∈ m X † . =-=(3)-=- if X ∈ exp(m)4 A new algorithm for producing quantum circuits using KAK decompositions k and exp(k) m Fig. 1. Patterns of an element of k, exp(k) and m for a three-qubit system, where each square re... |

2 |
Synthesis of quantum logic circuits”, quant-ph/0406176, to appear
- Shende, Bullock, et al.
(Show Context)
Citation Context ...lt that an arbitrary 2 n × 2 n unitary matrix is composed of O(n 2 4 n ) elementary gates [7]. And improvement to O(4 n ) elementary gates has been reported by Möttönen et al. [4] and by Shende et al =-=[6]-=-. On the other hand, Khaneja and Glaser provided another kind of decomposition [8], which was later named KGD. KGD lies within the framework of the G = KAK theorem (cf. [9], Theorem 8.6) in Lie group ... |