Quantum symmetries and Cartan decompositions in arbitrary dimensions

D’Alessandro, Domenico; Albertini, Francesca

doi:10.1088/1751-8113/40/10/013

Cited by 22 publications

(18 citation statements)

References 18 publications

(47 reference statements)

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“…Here our result is better than those of the former studies. By substituting formula (13) into formula (15), we get the sequence of control pulses and drifts to implement the gate in a circular three spins chain i 4 e 4 4 4 2…”

Section: The Implementation Of the Cnot Gates And The Three-qubit Swamentioning

confidence: 99%

“…The most widely used forms of Cartan decomposition in quantum information processing are Khaneja-Glaser decomposition [5], concurrence canonical decomposition [11,12], odd-even decomposition [13], and a kind of Cartan decomposition for bipartite quantum system in high dimension [14][15][16]. Based on Cartan decomposition, the problems of the synthesis, optimization and "small circuit" structure of two-qubit gate are completely solved [17][18][19][20].…”

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confidence: 99%

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Synthesis of some three-qubit gates and their implementation in a three spins system coupled with Ising interaction

Wei

Wang

2010

Sci. China Phys. Mech. Astron.

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The synthesis of the Toffoli gate, Fredkin gate, three-qubit Inversion-on-equality gate and D(α) gate, as well as their implementation in a three spins system coupled with Ising interaction are investigated. The sequences of the control pulse and the drift process to implement these gates are given. It is revealed that the implementation of some three-qubit gates in a circular spin chain is much better than in a linear spin chain, and every two measurements of the quantum computation complexity are not always consistent. It is significant to directly study the implementation of the multi-qubit gates and even more complicated components of quantum information processing without resorting to their synthesis. three-qubit gate, three spins system, Ising interaction, the sequences of control pulses and drift processes PACS: 03.67.Lx, 03.65.-Aa, 03.65.FdThe realization of quantum information requires the accurate control of quantum states [1,2]. The active control of quantum states is realized by decomposing the evolution unitary matrix of the system into products of several realizable matrices. That is equivalent to exerting a certain control field on the system, and the goal of control is usually achieved by a sequence of control pulses [3][4][5][6]. The quantum information processes, especially the quantum computation, are usually described by the quantum circuit model [1]. Quantum logic gates are basic elements of quantum circuits. In 1995, Barenco et al. showed the universality of the set of one-qubit gates and CNOT gates [7]. The process of constructing quantum circuits by these elementary gates is called synthesis by some authors. The complexity of the quantum information processing can be measured by the number of elementary gates to constitute the quantum circuits, and it can also be further measured by the number of the control pulses and drift processes.The decomposition of matrix plays a very important role in the implementation and optimization of quantum gates.The methods of decomposition currently used in this field mainly are Cartan decomposition [8] based on group theory, cosine-sine decomposition (CSD) [9] based on numerical linear algebra and quantum Shannon decomposition (QSD) [10] proposed by Shende, Bullock and Markov. The most widely used forms of Cartan decomposition in quantum information processing are Khaneja-Glaser decomposition [5], concurrence canonical decomposition [11,12], odd-even decomposition [13], and a kind of Cartan decomposition for bipartite quantum system in high dimension [14][15][16]. Based on Cartan decomposition, the problems of the synthesis, optimization and "small circuit" structure of two-qubit gate are completely solved [17][18][19][20]. Much progress is made in the exploration of the optimal elementary gate number required to synthesize a general n-qubit gate [10,19,21,22].The best result is that the number needs at most 23 4 48 n 3 4 2 2 3 n − + CNOT gates asymptotically, which is obtained by the QSD [10]. But for specific multi-qubit gates, our work reveals the opti...

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Section: The Implementation Of the Cnot Gates And The Three-qubit Swamentioning

confidence: 99%

mentioning

confidence: 99%

Synthesis of some three-qubit gates and their implementation in a three spins system coupled with Ising interaction

Wei

Wang

2010

Sci. China Phys. Mech. Astron.

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“…Khaneja-Glaser Decomposition (KGD) [8] . Moreover, there are some other decompositions, such as Concurrence Canonical Decomposition (CCD) [17,18] , which is a decomposition of (2 ) n SU group, too, and Odd-Even Decomposition (OED) [19] , which is the generalization of CCD to a more general multipartite quantum system case. A kind of Cartan decomposition for a bipartite quantum system was discussed in ref.…”

Section: Cartan Decomposition Of Lie Groupmentioning

confidence: 99%

Cartan decomposition of a two-qutrit gate

Zhang

Wei

2008

Sci. China Ser. G-Phys. Mech. Astron.

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The decomposition of matrices corresponding to the 2-qutrit logic gate by successive Cartan decomposition is investigated, and written in an exponential form based on the relationship between Lie group and Lie algebra, thus making them able to relate with the control field and the Hamiltonian of the system to perform the gate. Finally the decomposition of the ternary SWAP gate is presented in detail.Cartan decomposition, Cartan subalgebra, 2-qutrit system, ternary SWAP gate Rapid development of quantum information [1] and selected bond chemistry [2] requires accurate control on quantum states. The evolvement of a quantum system can be resolved into the time evolution unitary operator. As we know, quantum circuits, quantum logic gates, unitary operators and the corresponding unitary matrices have the same meaning in quantum information science. In order to achieve active control on quantum states, it is necessary to decompose the evolution unitary matrix into products of several realizable matrices [3,4] , which is equivalent to exerting a certain control field on the system. The control goal is usually gained by a sequence of control pulses [5] .Decomposition of the matrix plays a very important role in quantum control. In 1995, Barenco et al. showed the universality of the set of one-qubit gate and CNOT gate by use of QR decomposition of unitary matrices [6] . However, the traditional QR decomposition technique needs 3

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“…Typically, in the approach of quantum control, one should first model the controlled system and examine its controllability which is determined by the system Hamiltonian and interaction Hamiltonian with classical fields, and then design classical fields to stream the system to the given target state, which is referred to as the control protocol and is the issue we would like to address in this paper. Some works were proposed along this line, for example, using the Cartan decomposition of Lie groups [12].…”

Section: Introductionmentioning

confidence: 99%

Control Protocol of Finite Dimensional Quantum Systems

Tang¹,

2013

Commun. Theor. Phys.

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An exact and analytic control protocol of two types of finite dimensional quantum systems is proposed. The system can be drive to an arbitrary target state using cosine classical fields in finite cycles. The control parameters which are time periods of interaction between systems and control fields in each cycles are connected with the probability amplitudes of target states via triangular functions and can be determined analytically.

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Quantum symmetries and Cartan decompositions in arbitrary dimensions

Cited by 22 publications

References 18 publications

Synthesis of some three-qubit gates and their implementation in a three spins system coupled with Ising interaction

Synthesis of some three-qubit gates and their implementation in a three spins system coupled with Ising interaction

Cartan decomposition of a two-qutrit gate

Control Protocol of Finite Dimensional Quantum Systems

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