Realization of Two-Qutrit Quantum Gates with Control Pulses

Commun. Theor. Phys. Beijing, China 51 pp. 65 65 c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No., April 15, Realization of Two-Qutrit Quantum Gates with Control Pulses ZHANG Jie, DI Yao-Min, and WEI Hai-Rui School of Physics & Electronic Engineering, Xuzhou Normal University, Xuzhou 1116, China Received July, Abstract We investigate the realization of -qutrit logic gate in a bipartite -level system with qusi-ising interaction. On the basis of Cartan decomposition of matrices, the unitary matrices of -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 -qutrit logic gate is made first, and then the realization of the ternary SWAP gate and the ternary SWAP gate are discussed specifically, and the sequences of control pulses and drift processes implementing these gates are given. PACS numbers:.67.lx,.65.fd Key words: Cartan decomposition, -qutrit system, ternary SWAP gate, ternary SWAP gate 1 Introduction The realization of quantum information [1,] requires the accurate control on quantum states. The evolvement of a quantum system can resolve into the time evolution unitary operator. In order to achieve the active control on quantum states, it is necessary to decompose the evolution unitary matrix into products of several realizable matrices. That is equivalent to exerting a certain control field on the system, and the control goal is usually gained by a sequence of control pulses. [ 6] Much effort has been devoted to the construction of quantum circuits by use of qubit. [6] The current standard paradigm is based on a combination of quantum controlled-not CNOT gates between pairs of qubits and single-qubit gates. [61] The situation using the elementary gate other than CNOT gate is discussed in Refs. [1] [1]. The realization of n-qubit gate based on the Hamiltonian of the system and the control field has been discussed in Ref. [6]. Recent studies have indicated that there are many advantages to expand quantum computer from double-value system qubit to multi-value system qudit. The synthesis of multi-qutrit gate and multi-qudit quantum logic circuits have been discussed in Refs. [15] and [16]. In Ref. [15], a general -qutrit logic gate can be resolved into four 1-qutrit quantum multiplexers and three 1-qutrit uniformly controlled rotations acting on the first qutrit. But the realization of these two basic components needs study further. Decomposition of matrix plays a very important role in quantum control. A kind of Cartan decomposition for a bipartite quantum system was discussed in Ref. [17]. The Cartan decomposition for bipartite -level quantum system was discussed specifically in Ref. [1]. In this paper we devote to the discussion on the realization of general -qutrit gate based on the Hamiltonian and control field. In the decomposition of matrix, we will follow the method given in Ref. [1]. This paper is organized as follows. In Sec., on the basis of Cartan decomposition of matrices, the realization of -qutrit logic gate in a bipartite -level system with qusi- Ising interaction is generally investigated. The realization of the ternary SWAP gate and the ternary SWAP gate are specifically studied respectively in Secs. and, and the sequences of control pulses and drift processes implementing these gates are given. Finally, we give a brief conclusion in Sec. 5. Realization of Two-Qutrit Gates A qutrit gate is a operation acting on -level quantum system, whose Hilbert space is three-dimensional. We choose Hermitian matrices as following, σ 1 x = σ 1 z = σ y = σ 1 x = 1 1 1 i i 1 1, σ 1 y =, σ x =, σ z =, σ 1 y = i i, 1, 1 1, i, i The project supported by the National Natural Science Foundation of China under Grant No. 65 and the Science Foundation of Xuzhou Normal University under Grant No. 6XLA5 Corresponding author, E-mail: Yaomindi@sina.com

65 ZHANG Jie, DI Yao-Min, and WEI Hai-Rui Vol. 51 σ z 1 = 1. 1 In the nine σ matrices only eight matrices are independent and multiplying them by i we gain the basis vectors of Lie algebra su for one qutrit. Two qutrit is a bipartite -level quantum system, the Hamiltonian of the system may be written as H = H d + v i th i, i where H d is the part of Hamiltonian that is internal to the system and we call it the free evolution Hamiltonian and i v ith i is the part of Hamiltonian that can be externally changed called control Hamiltonian. The time evolution operator matrix satisfies Ut = ihtut, U = I. Since Ising interaction is widely used in quantum information science, such as to discuss the realization of multi-qubit gates [1 1] and the entanglement of multiqubit quantum system. [,] Now we assume the system has qusi-ising interaction, H d = Jσ 1 σ1 / Jσ z 1 σ z 1 /, which is a generalization of Ising interaction to qutrit case. We choose H i = σ λµ /, l = 1,, which corresponds to the hard pulses that excite each of the qutrits individually. Here we have σ λµ 1α = σ α λµ I, = I σλµ α, α = x, y, z. σ λµ α In Ref. [1], the Cartan decomposition of U matrix for -qutrit gate is given by X = K 1 A 1K A 1K A K AK 5 A K 6 A K 7 A K, where A, A j j = 1, and A j j = 1,,, belong to the Abel groups associated to Cartan subalgebras α, α, and α that appear in each decomposition step. They may be written in exponent form A = e α, A = e α, A = e α. 5 These Cartan subalgebras are α : = span{ii, iσ λµ lz, iσ λµ σ λ µ }, 6 α : = span{iσ y, iσ1 σ y, iσ σ y, iσ 1y + σ 1y σ σ 1y σ1 }, 7 α : = span{iσ σ y, iσ 1y σ x, iσ1 σ1 y σ 1 1y σ1 x }, where l = 1, λµ = 1, and λ µ = 1, in Eq. 6. The K j j = 1,,..., is conjugate to Lie group SO SO1 SO SO, which can be further decomposed. Although the matrix decomposition for -qutrit gate is much more complicate than that for - qubit gate, the U matrix X can be finally factorized into products of one parameter Lie groups, which are unitary transformations of the form or N λµ α = exp iασλµ α R, exp ibσ λµ 1α σλ µ α b R. 1 The qusi-ising interaction here can produce a transformation of the form Mb = exp ibσ 1 σ1. 11 Note that for R λµ= exp i σλµ, ly λµ= exp i σλµ ly, we have R 1y 1R y 1 exp ibσ 1 R σ1 R 1y 1R y 1 = exp ibσ 1 σ1 x, 1 1R x 1 exp ibσ1 σ1 R 1R x 1 = exp ibσ 1 1y σ1 y, 1 R 1y 1R x 1 exp ibσ1 σ1 R 1y 1R x1 = exp ibσ 1 σ1 y, 1 R 1R y1 exp ibσ 1 σ1 R 1R y 1 = exp ibσ 1 1y σ1 x, 15 1y 1 y 1 exp ibσ 1 σ1 1y 1 y 1 = exp ibσ σ, 16 1y y exp ibσ 1 σ1 1y y = exp ibσ 1 σ1, 17 1y 1 y expibσ 1 σ1 1y 1 y = exp ibσ σ1, 1 1y y 1 expibσ 1 σ1 1y y 1 = exp ibσ 1 σ, 1 and so on. By use of these relations, the other form of transformation in Eq. 1 can be obtained. N λµ a, R λµ and λµ can be realized by the control pulses, which correspond to Hamiltonian H i in Eq., and Mb, which corresponds to H d, is a free evolution of the system and we call it the drift. So -qutrit gates can be implemented by a sequence of pulses N λµ a, R λµ, ly λµ, and drift Mb. Realization of Ternary SWAP Gates It is a very complicate task to decompose the most general -qutrit gate which has 1 parameters. But for a specific ternary gate commonly used the situation can be much simpler. Let us concretely discuss the realiza-

No. Realization of Two-Qutrit Quantum Gates with Control Pulses 655 tion of the ternary SWAP gate first. Analogous to the binary SWAP gate, the action of the ternary SWAP gate is defined as X sw : i 1 j j 1 i, where i, j =, 1, and {, 1, } 1, are orthonormal basis for the Hilbert space of the subsystems. This operator is relevant in quantum information and computation since it enables to switch the quantum states of different system. The matrix representation of this gate is given by X sw = 1 1 1 1 1 1 1 1 1. 1 The Cartan decomposition of X sw has been given in Ref. [1], that is, X sw = KA A = L 1 L A A. Each factor in Eq. can be written as following exponent form L 1 = e l 1, L = e l, A = e α, A = e α. The exponent is expressed in the bases of u algebra as follows, l 1 = i σ,1 σ,1 y, l = i σ,1 1y σ,1 x, 5 α = i σ, 1y σ, + σ 1, 1y σ 1, x σ, x σ 1, σ, y σ1, y, 6 α = i I σ,1 + σ, + σ,1 σ, σ,1 σ,1 σ, σ, + σ, σ,1. 7 A and A belong to Abel group, they can be directly factorized into products of one parameter subgroups, while L 1 and L belong to one parameter groups. By use of the relations in Eqs. 1 1 and similar transforms, we get L 1 = R 1y 1Rx 1M R1y 1R x1, L = R 1R y1m A = R 1y R x 1y 1 y 1M R 1Rx 1, 1y 1 y 1R 1y R xr R y 1y 1 y 1M 1y 1 y 1R Ry R 1y1Rx 1 1y y M 1y y R 1y 1R x1 R 1R y1 1y y M 1y 1, y R 1Ry A = e i/ N 1 N N 1 N M 1y 1 y 1M 1y 1M 1y 1. 1 According to Eqs. and 1, the ternary SWAP gate can be written as products of a series of realizable matrices. Since the operators acting on different subsystem commute, moreover, we have the relations as follows: 1y 1R 1y R 1y 1 = R1y 1R 1, y 1R xr y y 1 = R x 1R y 1, 1y 1R R 1y 1 1y = R 1R1y 1 1y, y 1R y R x 1 y = R y 1R x y 1, 5 1y R 1y 1R 1 1y = R 1y 1R 1, 6 y R x1r y 1 y = R x 1Ry 1, 7 N 1 N = N 1, N 1 N = N 1, the expression of X sw is finally written as: R1y 1R 1 R x 1R y 1M R 1R 1y 1y 1 Ry 1R x y 1M R1y 1 X sw = e i/ R 1y 1R x 1M R 1R x1r y 1M R1y 1 1y R y 1R x R 1y 1R Ry 1M R 1 1y 1 N 1 1R x1 1y y R y1 y 1M 1y 1 N 1 R 1 y 1.

656 ZHANG Jie, DI Yao-Min, and WEI Hai-Rui Vol. 51 Implementing the sequences of control pulses R λµ, ly λµ and N λµ a and drift processes Mb described in Eq. 7, the ternary SWAP gate can be obtained. Realization of Ternary SWAP Gates The binary SWAP gate is an important logic gate for quantum computation, moreover it is a perfect entangler for it s fine entanglement property. [,5] Now we generalize this gate to ternary case. Assume ternary SWAP gate X sw = X swx sw, where detx sw = 1, det X sw =, and X sw = X swd, D = diagi,, I 5, so we define two kinds of ternary SWAP gate X sw and X sw. The matrix representation of X sw is given by X sw = 1 / / 1 + i/ 1 i/ / / 1 1 + i/ 1 i/ 1 i/ 1 + i/ 1 i/ 1 + i/ 1. 1 Now we carry out the Cartan decomposition of X sw by use of the method in Ref. [17]. The results are X sw = KA 1AA, X sw = KA. A and à belong to the Abel group associated to Cartan subalgebra appearing in the first step of decomposition U/SO decomposition, the A 1 and A belong to the Abel group associated to Cartan subalgebra appearing in the third step of decomposition the further decomposition of SO5 SO, while the K is conjugate to Lie group SO SO1 SO SO. The matrix expressions of these components are A = diag{i, i,, 1, i, I }, à = diag{i, i, I, i, I }, 5 K = diag 1, 1, I 5, 6 1 1 A 1 = diag I,, 1, 7 and A is the transpose of A 1. Each factor in Eq. to Eq. can be written in exponent form as follows, K = e k, A 1 = e α 1, A = e α, A = e α, à = e α, where k = i σ,1 σ,1 y σ,1 1y σ,1 x, α 1 = i σ, 1y σ, x σ, + σ 1, 1y σ 1, x σ 1, α = i 6 σ,1 + σ,1 σ,1 σ1, σ, y y, 5 i σ, + σ, σ, σ,1 σ,, 51 α = i I 1 σ,1 + σ, + σ,1 σ, 1 σ,1 σ,1 σ, σ, + σ,1 σ, + σ, σ,1, 5 α = α 1. 5 Then we gate K = R 1y 1Rx 1M R1y 1R x1 R 1R y1m R 1Ry 1, 5 A 1 = R 1y Rx 1y 1 y 1M 1y 1 y 1R 1y R xr R y 1y 1 y 1M 1y 1 y 1R Ry R 1y1Rx 1 1y y 1y R y 1 1y y M A = N 1 y R 1Ry N M 6 y R 1y 1R x1r 1 1y 1, 55 6 1y 1 y 1

No. Realization of Two-Qutrit Quantum Gates with Control Pulses 657 1y 1M Ã = e i/ N 1 N y 1, 56 N 1 1 N M 1y 1M y 1 1 1y 1M y 1, 57 A = R 1y R x 1y 1 y 1M 1y 1 y 1R 1y R xr R y 1y 1 y 1M 1y 1 y 1 R Ry R 1y1Rx 1 1y y M 1y y R 1y 1 R x 1R 1R y1 1y y M 1y y R 1Ry 1. 5 Analogous to the discussion on the ternary SWAP gate X sw, the X sw and X sw can be finally expressed as follows, X sw = R 1y 1Rx 1M R1y 1R 1 R x 1R y 1M R 1R 1y 1y 1R y 1R x y 1M R 1y 1R 1R x1r y 1M R 1R1y 1 1y R y 1R x y 1M R 1y 1R 1R x1 Ry 1M 1y R 1 1y 1 y R y1m 1y 1 y 1M N 1 N 6 6 R 1y 1y 1Rx y 1M R1y 1R 1R x1r y 1M R 1R1y 1 1y R y 1R x y 1M R 1y 1R 1 R x 1Ry 1M 1y R 1 y R 1 ; 5 y X sw = e i/ R 1y 1R x 1M R 1R x1r y 1M R 1y 1 R 1 R 1y 1y 1Ry 1R x y 1 R1y 1R 1R x1r y 1 R 1R1y 1 1y R y 1 R x y 1M R 1y 1R 1 R x 1Ry 1M 1y R 1 y R y1m 1y 1 y 1M 1y 1 N 1 1 N 1 N 1 N R 1y 1y 1 Rx y 1M R1y 1R 1 R x 1R y 1M R 1R 1y R y 1R x y 1M 1y 1 R 1y 1R 1R x1r y 1M 1y R 1 y R y 1. 6 So we get the sequences of control pulses and drift processes required to implement two kinds of SWAP gate. 5 Conclusions From the discussion above, we can see that although the Cartan decomposition of -qutrit is much more complicated than that in -qubit case, the -qutrit gate can still be realized by implementing sequences of control pulses and drift processes. The -qutrit system can carry much more information than that in -qubit case. The -qutrit gate corresponds to U matrix, while -qubit U matrix. Moreover the binary SWAP gate requires CNOT gates [] and each CNOT gate needs control pulses and 1 drift processes. [6] From this consideration, although the number of control pulses and drift is rather large, the results get here for ternary SWAP gate and ternary SWAP gates is reasonable and rather good. References [1] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University, Cambridge. [] H. Rabitz, R.D. Vivie-Riedle, M. Motzkus, and K. Kompa, Science. [] S.G. Schirmer, A.D. Greentree, V. Ramakrishna, and H. Rabitz, J. Phys. A 5 15. [] S.G. Schirmer, A.D. Greentree, V. Ramakrishna, and H.

65 ZHANG Jie, DI Yao-Min, and WEI Hai-Rui Vol. 51 Rabitz, quant-ph/15155. [5] D. Bonacci, S.D. Bosannac, and N. Došlić, Phys. Rev. A 7 1. [6] N. Khaneja and S.J. Glaser, Chem. Phys. 67 1 11. [7] A. Barenco, et al., Phys. Rev. A 5 15 57. [] F. Vatan and C.P. Williams, Phys. Rev. A 6 15. [] F. Vatan and C.P. Williams, quant-ph/117v. [1] M. Möttönen, J.J. Vartiainen, V. Bergholm, and M.M. Salomaa, Phys. Rev. Lett. 15. [11] Y. Liu, G.L. Long, and Y. Sun, arxiv abs/7.7. [1] J. Zhang, J. Vala, S. Sastry, and K.B. Whaley, Phys. Rev. Lett. 5. [1] J. Zhang, J. Vala, S. Sastry, and K.B. Whaley, Phys. Rev. A 6. [1] Y.S. Zhang, M.Y. Ye, and G.C. Guo, Phys. Rev. A 71 5 61. [15] F.S. Khan and M.M. Perkowski, Proceedings of the 7th International Symposium on Representations and Methodology of Future Computing Technologies, Tokyo, Japan 5. [16] F.S. Khan and M. Perkowski, Theor. Comput. Sci. 67 6 6. [17] D. D Alessandro and R. Romano, J. Math. Phys. 7 6 1. [1] Y.M. Di, J. Zhang, and H.R. Wei, Science in China Series G 51 166. [1] J.A. Jones, Phys. Rev. A 67 117. [] M.D. Bowdrey, J.A. Jones, E. Knill, and R. Laflamme, Phys. Rev. A 7 5 15. [1] M. Van den Nest, W. Dür, and H.J. Briegel, Phys. Rev. Lett. 1 1151. [] W. Dür, L. Hartmann, M. Hein, M. Lewenstein, and H.J. Briegel, Phys. Rev. Lett. 5 7. [] A. Lakshminarayan and V. Subrahmanyam, Phys. Rev. A 71 5 6. [] X.G. Wang and B.C. Sanders, Phys. Rev. A 6 11. [5] B.L. Hu and Y.M. Di, Commun. Theor. Phys. Beijing, China 7 7 1. [6] J. Kim, J.S. Lee, and S. Lee, Phys. Rev. A 61 1.