Numerical simulation of leakage effect for quantum NOT operation on three-josephson-junction flux qubit

Transcription

1 Numerical simulation of leakage effect for quantum NOT oeration on three-josehson-junction flux qubit 1 Tao Wu, 1 Jianshe Liu, 2 Zheng Li 1 Institute of Microelectronics, Tsinghua University, Beijing Deartment of Electronic Engineering, Tsinghua University, Beijing 184 Suerconducting flux qubits with three Josehson junctions are romising candidates for the building blocks of a quantum comuter. We have alied the imaginary time evolution method to study the model of this qubit accurately by calculating its wave functions and eigenenergies. Because such qubits are maniulated with magnetic flux microwave ulses they might be irradiated into non-comutational states which is called the leakage effect. Through the evolution of the density matrix of the qubit under either hard-shaed π-ulse or Gaussian-shaed π-ulse to carry out quantum NOT oeration,it has been demonstrated that the leakage effect for a flux qubit is very small even for hard-shaed microwave ulses while Gaussian-shaed ulses may suress the leakage effect to a negligible level. PACS number(s): r, 3.67.Lx, C Suerconducting qubits [1, 2] are solid-state macroscoic systems conforming to the rinciles of quantum mechanics at ultra low temeratures. They can be comatibly fabricated in a microelectronic rocess line and easily maniulated by on-chi microwave currents or flux ulses, which make them romising candidates for the building blocks of a quantum comuter. [3] For suerconducting flux qubits, [4-9] the magnetic flux is the convenient arameter to mark and control their eigenstates. Flux qubits are insensitive to background charge fluctuations but relatively fragile to magnetic flux noise comared to suerconducting charge [1, 11] and hase [12, 13] qubits. Quantum suerosition in the sectroscoy [6] and Rabi oscillations of a flux qubit in the time domain [7] have been observed. Leakage effect [13-15] of a flux qubit during quantum oerations means that the qubit escaes into non-comutational subsaces. This effect of hase qubits has been discussed, [13, 14] and insired by Lin in Ref. [15] we numerically study it for a flux qubit in this letter. This work is based uon the calculation of the eigenstates of a single flux qubit using the imaginary time evolution method [16-19] and the evolution of its density matrix under magnetic flux ulse erturbation. It has been revealed that the oulations irradiated to the second and third excited states are extremely small for Gaussian-shaed ulses and also unimortant for hard-shaed ulses, thus there is no need to consider other excited states because the leakage to higher energy levels are much smaller and therefore negligible

2 A three-josehson-junction (3JJ) flux qubit is comosed of a suerconducting loo interruted by three Josehson junctions [4-6] as shown in Fig.1s. Junctions 1 and 2 have equal areas while junction 3 is α (<α<1) times smaller. The critical currents for them are I c, I c and αi c, resectively. The loo is biased by a magnetic flux fφ, where f is the flux frustration, Φ =h/(2e) is the suerconducting flux quantum, e is the electron charge, and h is Planck costant. The qubit can be reduced to a two-level system when f is in the vicinity of n+1/2 with n an integer. Neglecting the loo inductance, the Hamiltonian of the system can be written as [5] 1 rt 1 r H = M + EJ [2 + α cos( ϕ1) cos( ϕ2) αcos(2 π f + ϕ1 ϕ2)], (1) where Φ α α + r r T T M= C j, = Mϕ&, i.e., ( 1 2) = M ( & ϕ & 1 ϕ2) and 2π α 1+ α E /(2 ) J = IcΦ π is the Josehson energy for junction 1 and 2. Eq. (1) can be exressed exlicitly as 2 + α) π 2 2 2α 1 2 ( 1 2 ) + α ) C Φ 1+ α + EJ α ϕ1 ϕ2 α π f ϕ1 ϕ2 (1 2 H = + + 2(1 2 J [2 + cos( ) cos( ) cos(2 + )]. (2) It is different from the usual one as in Ref. 5 because the Hamiltonian remains in the original frame without transformation. We have utilized the fourth order imaginary time evolution method [16-19] to calculate the above Hamiltonian. TableⅠshows the eigen-energies of the qubit for f =.5 and f =.495. Figs. 2A-B illustrate the ground state and first excited state 1 for f =.5, Figs. 2C-D show and 1 for f=.495, and Figs. 2E-F show the second excited state 2 and third excited state 3 for f=.495. We choose in the comutation α=.8, E J = GHz and C J =7.765 ff. Table Ⅰ. The energies E i for the lowest four eigenstates with two values of f. The energies are in units of GHz. f E E 1 E 2 E The wave functions of and 1 for f=.5 are symmetric and anti-symmetric, resectively, and they quickly lose the symmetry when f deviates from this degenerate oint. To achieve larger suercurrents for readout, we bias the qubit at f=.495, where the wave functions of and 1 aear localized in two searate wells as shown in Figs. 2C and 2D. Based on these results, we have studied the leakage effect during a quantum NOT oeration uon one flux qubit with microwave magnetic flux ulse oerations. The interaction term W(t) between the microwave field and the qubit can be considered as a erturbation on the original Hamiltonian in Eq. (2), or an external microwave magnetic flux erturbation fµ ()cos t ( ωt) Φon the flux bias fφ threading the loo, - 2 -

3 i.e., W() t 2παE f ( t) cos( ω t) sin( 2π f + ϕ ϕ ), (3) J µ 1 2 where fµ ( t) << f and ω = ( E1 E )/ h. From now on, we denote by h ω the energy unit. In the Hilbert sace saned by the eigenstates, 1, 2 and 3, the interaction W(t) can be written as a 4 4 matrix: W = F( ) M cos( ), where = ω t /(2 π), F( ) = 2πα EJ f µ ( ), and M is a 4 4 matrix with Mi, j= i 1sin( 2π f + ϕ1 ϕ2) j 1 for i, j=1,2,3,4. Note that F() is controllable in exeriments. In the comutational sace, the oulation inversion between the ground state and the first excited state deends on the effective matrix element M 1,2 and M 2,1, i.e.,.339. A quantum NOT oeration can be achieved by a microwave π-ulse with duration, which requires ( ) 1,2 F M d = 1/2. (4) The hard-shaed ulse has constant microwave amlitude, and Eq. (4) yields F( ) = 1/(2 M1,2) for.for the Gaussian-shaed ulse, [15,16] one has 2 F( )=( 1/ 2M1,2 )( 1/ 2πw) ex( ( / 2) / 2 w) for, where w is the characteristic width of the Gaussian ulse chosen between.167 and.1. [2] The evolution of the qubit under irradiation can be described in the interaction icture by the Liouville equation: [21] 1 ( ) i ρ( ) / = 2 π[ H, ρ( )]. (5) Here H1 ( ) is a matrix with ( ) ex, ( 2 π( ) i j ) k ( E E )/ ( ω ) H = i k k W, where 1 i j i, j 3 i = i 1 h. Choosing a time ste small enough (e.g., 1 ), we can integrate the equation to obtain the density matrix for a given time. The element ρ at the end denotes the robability P i of the system in state i, i=1, 2, 3, 4. Table ii, Ⅱshows the final values of the leaking oulations for three time durations of ulses. Fig. 3 illustrates the leakage effect during the oerations for =4 (i.e., 45.5ns). Table Ⅱ. The oulations (P i ) of the lowest four levels after the microwave magnetic flux irradiations with Hard-shaed ulses (on the left of / ) and with Gaussian-shaed ulses (on the right of / ). =1 =2 =4 P 2 1.3e-5/4.6e-8 2.5e-6/1.4e-8 1.5e-6/4.2e-9 P 3 2.9e-6/5.e-8 1.e-6/1.e-8 3.3e-7/1.8e-9 It is revealed that Gaussian-shaed π-ulse inhibit leakage effects 1 better than hard-shaed π-ulse, which is as remarkable as that in the case of hase qubits. [13] Also, longer durations reduce the leakage. However, leakage is very small during this - 3 -

4 oeration even for hard-shaed ulses. In conclusion, we have solved the eigen-functions and eigen-energies of a 3JJ flux qubit by the imaginary time evolution method and studied the leakage effect during the quantum NOT oeration through the evolution of its density matrix. It has been demonstrated that the leakage effect for a flux qubit is very small even for hard-shaed microwave ulses while Gaussian-shaed ulses may suress the leakage effect to a negligible level. We gratefully areciate Johnson P R and Strauch F W in NIST for their instructions about the imaginary time evolution method. Discussions with Wang Ji-lin, Li Tie-fu, Chen Pei-yi, Yu Zhi-ing and Li Zhi-jian are acknowledged. This work is suorted by the 211 Program of Nanoelectronics of Tsinghua University (21651). References [1] Makhlin Yu, Schön G and Shnirman A 21 Rev. Mod. Phys. 73, 357 [2] Devoret M H and Martinis J M 24 Quantum Information Processing 3, 163. [3] Nielsen M A and Chuang I L 2 Quantum Comutation and Quantum Information (Cambridge: Cambridge University Press, UK) [4] Mooij J E et al Science 285, 136 [5] Orlando T P et al Phys. Rev. B 6, [6] van der Wal C H et al. 2 Science 29, 773 [7] Chiorescu I, Nakamura Y, Harmans C J and Mooij J E 23 Science 299, 1869 [8] Greenberg Y S et al. 22 Phys. Rev. B 66, ; 22 ibid 66, ; 23 ibid 68, [9] You J Q, Nakamura Y and Nori Franco 25 Phys. Rev. B 71, [1] Nakamura Y, Pashkin Y A and Tsai J S 1999 Nature 398, 768 [11] Vion D et al. 22 Science 296, 886 [12] Martinis J M, Nam S, Aumentado J and Urbina C 22 Phys. Rev. Lett. 89, [13] Steffen M, Martinis J M and Chuang I L 23 Phys. Rev. B 68, [14] Amin M H, cond-mat/478 [15] Tian L, Ph. D. thesis 22 MIT [16] Johnson P R et al. 23 Phys. Rev. B 67, 259 [17] Auer J, Krotscheck E and Chin S A 21 J. Chem. Phys. 115, 6841; Krotscheck E et al. 23 International Journal of Modern Physics B 17, 5459 [18] Takahashi K and Ikeda K 1993 J. Chem. Phys. 99, 868 [19] Feit M D, Flek J A, Jr. and Steiger A 1982 J. Comut. Phys. 47, 412 [2] This choice is based uon the aroximation 2 2 ( ) ex ( / 2) / 2 d 2π w w - 4 -

5 with = k w and k 6 1. [21] Blum Karl 1981 Density Matrix Theory and Alications (NewYork: Plenum Press) Figure Cations Fig. 1. One flux qubit with crosses reresenting Josehson junctions. C J1, C J2 and C J3 are the equivalent caacitances of the junctions with C J1 =C J2 =C J and C J3 =αc J. Fig. 2. (A-B)Ground state and first excited state 1 for f=.5. (C-F), 1, second excited state 2 and third excited state 3 for f =.495. Fig. 3 Poulation of (P, solid line), 1 (P 1,dashed line), 2 (P 2, solid line) and 3 (P 3, dashed line) after (A-B) the hard-shaed π-ulse and (C-D) the Gaussian-shaed π-ulse. The ulse duration is =4. Fig. 1 Fig

6 Fig