Quantum computing in decoherence-free subspaces with coupled charge qubits

ARTICLE IN PRESS

Physica E 40 (2008) 878–882 www.elsevier.com/locate/physe

Quantum computing in decoherence-free subspaces with coupled charge qubits Zhi-Bo Fenga,b,, Xin-Ding Zhangb a

National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, China Institute for Condensed Matter Physics, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510631, China

b

Received 3 September 2007; accepted 22 October 2007 Available online 9 November 2007

Abstract We present a simple but feasible quantum computing scheme in decoherence-free subspace by coupling four identical superconducting Josephson circuits. Two physical charge qubits are encoded to a logical one by connecting them with a common superconducting quantum interference device (SQUID). Well-controllable coupling between two logic qubits is further proposed by using a variable electrostatic transformer to construct the two-qubits controlled-phase gate. Taking into account the symmetry of system-bath interaction, our scheme may be helpful to suppress collective noises. r 2007 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 03.67.Pp; 85.25.Cp Keywords: Quantum computing; Superconducting charge qubit; Decoherence-free subspace

Towards practical quantum computation (QC), devices of superconducting Josephson circuits (DSJC) behaved as artificial quantum two-level systems have many advantages, such as convenient control, flexible design and accurate readout [1–4]. A number of schemes concerning multi-qubit operations with DSJC have been reported [5–8]. To further implement scalable quantum computers, high fidelity quantum operations, especially the two-qubit gate operations are preferred. One of the critical problems we are facing is how to couple two individual qubits efficiently. Some significant proposals of realizing controllable coupling between superconducting Cooper-pair boxes have been put forward, including inductance coupling [9–11], junction coupling [12], LC-resonator coupling [13], etc. In particular, the coupling of charge qubits can be achieved through a shared dc-SQUID device [14,15]. Recently, a novel method to couple two charge qubits by a variable electrostatic transformer was presented Corresponding author at. National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, China. E-mail address: [email protected] (Z.-B. Feng).

1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.10.086

theoretically [16] and experimentally realized in Ref. [17]. Despite the progress achieved, more efforts are required to implement multi-qubit operations with DSJC based on well-controlled interqubit coupling. With the steps heading for practical, scalable quantum computing, decoherence effects during quantum operations have attracted increasing attention. As is well known, quantum coherence is fragile, especially for the solid quantum systems such as superconducting charge qubits (SCQs). Decoherence caused by the system–environment interaction may seriously destruct quantum operations. Fortunately some available schemes have been proposed to tackle the problem, e.g., quantum error correcting/avoiding codes [18–20] and geometric QC [21–24], etc. Furthermore, depending on the symmetry of system-bath interaction, one can choose a sort of decoherence-free subspaces (DFS) to implement QC [18–20,25,26]. The DFS idea has been experimentally tested in Ref. [27], and generalized to noiseless subsystems in Ref. [28]. For SCQs, the charge noises are mainly the Ohmic noise originated from external control circuits, and the background charge fluctuations [29–31]. Here we attempt to construct a set of universal quantum gates in a

ARTICLE IN PRESS Z.-B. Feng, X.-D. Zhang / Physica E 40 (2008) 878–882

well-designed DFS which is immune to the collective decoherence effects. In detail, two physical SCQs are coupled by a common SQUID to form a decoherence-free subsystem, which acts as a logical qubit. The two-logicqubit gate is realized through coupling together two subsystems with a variable electrostatic transformer. The qualitative analysis given out in the last place shows that our scheme may be helpful for isolating quantum operations from some major decoherence resources. 1. Dynamics of the coupled Cooper-pair boxes See Fig. 1(a), the system we investigate consists of four identical SCQs (signed as k ¼ 1; 2; 3 and 4). For each SCQ, a superconducting island (Cooper-pair box) is coupled to a ring by two symmetric Josephson junctions characterized by coupling energy E J0 and capacitance C J . Gate voltage V gk is applied to the kth box through the capacitance C k , then the charging energy of Cooper-pairs is changed by V gk inducing gate charges. The flux Fk threading the loop of the kth SQUID can modulate the Josephson coupling energy. Each two qubits 1 and 2 (3 and 4) are connected with a dc SQUID to form the subblock-a (b) [14,15]. The SQUID-m is pierced by a magnetic flux Fm , which provides a tunable Josephson coupling E Jm ¼ 2E ð0Þ Jm cosðpFm =F0 Þ,

(1)

where F0 ¼ h=2e is the fluxon, m ¼ a; b. Choosing small junction capacitances of SQUID-a (b), we can obtain that the electrostatic energy between boxes 1 and 2 (3 and 4) is much smaller than the corresponding Josephson energy [14]. In this case, the effect of electrostatic coupling energy can be ignored. To obtain a controlled gate operation, as

a (0)

EJ0 CJ

Φ1

(0)

Cm

Φ2

Φ3

Φa

EJb

Φ4

Φb

C1

C3

C2

Vg1

Vg 2

Cm2

Vg3

Ec

discussed in more details below, two subblocks a and b are coupled by a variable electrostatic transformer C m [16]. Since the system is operated in the charging regime, the extra Cooper-pairs nk in the kth box is a good quantum number, which acts as qubit jnk i. Near the charging energy degeneracy point ðngk ¼ 12Þ, only two charge states, say nk ¼ 0 and 1, play a dominant role [1]. Within the charge eigenbasis fj1i, j0igðkÞ , the system Hamiltonian in spin-12 representation reads [14–16]  X  E Jk ðkÞ ðkÞ ^ s^ H¼ k s^ z  2 x k¼1;2;3;4 E Ja ^ ðaÞ E Jb ^ ðbÞ ^ ð3Þ R  R þ Gs^ ð2Þ ð2Þ z s z . 2 x 2 x Here the charging energies are k ¼ E ck ð1  2ngk Þ=2 for k ¼ 1 and 4, where the charging energy scale is E ck ¼ 2e2 =C Sk , with total capacitance of the kth box C Sk ¼ C k þ 2C J , and ngk ¼ C k V gk =2e is the corresponding gate charge. However, for the qubits 2 and 3, due to the modification of the charging energy dcm [16] of the variable transformer C m , the charging energies become 

k ¼ E ck ð1  2ngk Þ=2 þ dcm ,

(3)

where the total capacitances are then C Sk ¼ C k þ 2C J þ C mk , with C mk being one part of C m (see Fig. 1(b)). Assuming that Fig. 1(b) has a symmetric structure, c ¼ C mk =C Sk for k ¼ 2 and 3, we have dcm ¼ ðDm =2Þ cosð2pq0 Þ cosð2pcÞ,

(4)

where Dm is the characteristic energy gap of the transformer junction, and the induced charge is  X 1 q0 ¼ qc þ c ngk  , (5) 2 k P with qc ¼ ðV c =2eÞ k C mk ð1  cÞ. The effective Josephson coupling energies are E Jk ¼ 2E J0 cosðpFk =F0 Þ,

b

EJa

879

C4 Vg4

Cm3 EJ

k ¼ 1; 2; 3 and 4. The average phase difference across two Josephson junctions of the kth SQUID is denoted by jk , which is canonically conjugate to nk , i.e., ½jk , nk  ¼ i. ðkÞ ^ ðkÞ Pauli operators s^ ðkÞ ¼ j1iðkÞ and p ðp ¼ z, xÞ satisfy s z j1i ðkÞ ðkÞ ðkÞ s^ ðkÞ ¼ j0iðkÞ , s^ ðkÞ ¼ j0iðkÞ and s^ ðkÞ ¼ j1iðkÞ . z j0i x j1i x j0i For SQUID-a (b), average phase difference ja ðjb Þ is relevant with extra Cooper-pairs n1 and n2 ðn3 and n4 Þ, this yields

cos ja ¼ ðjn1 ; n2 þ 1ihn1 þ 1; n2 j þ h:c:Þ=2. An analogous relation can be written for jb . Therefore, the ðaÞ ^ ð2Þ þ s^ ð1Þ s^ ð2Þ above operator is R^ ¼ s^ ð1Þ þ s þ , with inversion x

Vc

Fig. 1. (a) Schematic diagram of the coupled charge qubits, which consists of four identical SCQs. The qubits 1 and 2 (3 and 4) are coupled by the SQUID-a (b) to construct the subblock-a (b), respectively. Through a variable electrostatic transformer C m , the inter-subblock coupling can be switchable. (b) The equivalent circuit of C m proposed in Ref. [16].

(6)





ðkÞ ðkÞ ^ ðkÞ ^ ðkÞ operators s^ ðkÞ þ ¼ j1i h0j and s  ¼ j1i h0j, or s  ¼ ðkÞ ðkÞ ðs^ x  is^ y Þ=2. The last term G is the coupling strength between the two subsystems, in the tight-binding limit E c 5E J , it takes the form [16]

G¼

Dm cosð2pq0 Þ½1  cosð2pcÞ. 2

(7)

ARTICLE IN PRESS Z.-B. Feng, X.-D. Zhang / Physica E 40 (2008) 878–882

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E c and E J are the charging and Josephson coupling energies of the transformer C m . 2. Quantum gate operations in DFS Based on the symmetry of the system-bath interaction, we may choose a smaller space to encode logic qubits from the state space of an isolated subsystem. For example, in the subsystem-a, the charge eigenbasis span a space fj11i12 , j10i12 , j01i12 , j00i12 g. When the system-bath interaction ^ where Z^ ¼ s^ ð1Þ þ s^ ð2Þ , B^ is a has the form of Z^  B, z z 12 ^ random bath operator, there exist ðZ^  BÞj10i ¼ 0 and 12 12 12 ^ ^ ^ ^ ðZ  BÞj01i ¼ 0, since Zj10i ¼ 0 and Zj01i ¼ 0[18,19]. We consider the subspace C a :¼ spanfj10i12 ; j01i12 g,

eðC 2 þ 2C J þ C m2 Þ . 2ðC 2 þ 2C J ÞðC m2 þ C m3 Þ

Secondly, if the mutual inductance between the two subsystems is very small, the magnetic interaction between subsystem-a and b can be negligible. From Eq. (2), the Hamiltonian of the decoupled subsystem-a is given by  X E Jk ðkÞ E Ja ^ ðaÞ ðaÞ ^ H^ ¼ s R . k s^ ðkÞ  (8)  z x 2 2 x k¼1;2 Note that the charging energy is 2 ¼ E c2 ð1  2ng2 Þ=2 under the decoupled condition of q0 ¼ 14. The controllability of the isolated subsystem provides us the possibility of constructing the single-logic gates. The first goal is to derive the single-logic-qubit gate U z . From Eqs. (1) and (6), when Fa;1;2 are all equal to F0 =2, the intraqubit and interqubit couplings are E Ja;1;2 ¼ 0. Additionally, we appropriately choose the gate voltages V g1;2 to get 1 ¼ 2 . Hence, the system Hamiltonian is described ðaÞ ðaÞ by H^ ¼ 1 ðs^ ð1Þ  s^ ð2Þ Þ. We define R^ ¼ ðs^ ð1Þ  s^ ð2Þ Þ=2 z

z

ðaÞ

ðaÞ U z ¼ expðifz R^ z Þ,

(9)

where fz ¼ 21 tz =_ is the rotation angle around the z-axis. Next we demonstrate the logic-qubit gate U x through tuning appropriate parameters. When the applied fluxes Fa aF0 =2 and F1;2 ¼ F0 =2, the intraqubit and interqubit coupling energies are, respectively, E Ja a0 and E J1;2 ¼ 0. Furthermore, when the gate voltages V g1;2 are modulated to the degeneracy point ðng1;2 ¼ 12Þ, the charging energies will be 1;2 ¼ 0. As a result, the subsystem-a Hamiltonian is ðaÞ ðaÞ reduced to H^ x ¼ E Ja R^ x =2, which only remains the interqubit coupling. The corresponding evolution operator ðaÞ is U x ¼ expðiH^ tx =_Þ, with the evolution time tx . Since x

and define j1iaL ¼ j10i12 and j0iaL ¼ j01i12 as logic qubits. ^ For a random state jci 2 C a , ðZ^  BÞjci ¼ 0, which means that DFS C a is immune to the sz -type common bath. In the followings, we show in detail how to physically realize two single-logic gates U z and U x in the DFS. First, the coupling between the two subsystems should be switched off. In the light of Eqs. (5) and (7), the inter-subsystem coupling can be controlled by q0 induced by the voltages V g2;3 and V c . When Dm is determined and c is not an integer, the inter-subsystem coupling will be vanished, G ¼ 0 as long as q0 ¼ 14. Otherwise, the coupling is switched on. As a special example, if the gate voltages V g2;3 work at the degeneracy point, we have q0 ¼ 14 when Vc ¼

on j1iaL and j0iaL in the DFS C a can be achieved

z

z

ðaÞ

z

z

satisfying R^ z j1iaL ¼ j1iaL and R^ z j0iaL ¼ j0iaL . In the logic-quibt state space fj1iaL ,j0iaL g, the unitary operator is ðaÞ U z ¼ expðiH^ z tz =_Þ, with tz being the evolution time. After the operation, logic-qubits will acquire the corresponding phases fz , and then the rotation gate operation

ðaÞ ðaÞ R^ x j1iaL ¼ j0iaL and R^ x j0iaL ¼ j1iaL , the rotation gate U x in DFS C a as ðaÞ U x ¼ expðifx R^ x Þ,

(10)

can be constructed, fx ¼ E Ja tx =2_ is the rotation angle around the x-axis. Finally, we need to construct a nontrivial two-logicqubit gate, which is usually important and crucial for QC. The initially isolated subsystems a and b can be coupled by adjusting G from 0 to appropriate coupling strength Ga [16,17]. In coupled system, since the subsystem-b has a similar basis fj1ibL , j0ibL g, two-logic-qubits span a space C ab :¼ spanfj11iLL ; j10iLL ; j01iLL ; j00iLL g, here the superscripts have been neglected. By setting the fluxes Fk and Fa;b to F0 =2, we have E Jk ¼ 0 (k ¼ 1, 2, 3, 4) and E Ja;b ¼ 0. The coupled system Hamiltonian equation (2) will be X ^ ð3Þ H^ cp ¼ Gs^ ð2Þ k s^ ðkÞ z s z þ z . k¼1;2;3;4

Without loss of generality, the charging energies k ^ ð3Þ can be remained here. Since s^ ð2Þ z s z j11iLL ¼ j11iLL , ð2Þ ð3Þ ð2Þ ð3Þ ^ ð3Þ s^ z s^ z j10iLL ¼ j10iLL , s^ z s^ z j01iLL ¼ j01iLL , s^ ð2Þ z s z j00iLL ¼ j00iLL , we further obtain the unitary evolution operator U^ cp ¼ expðiH^ cp tcp =_Þ in the DFS C ab , with tcp being the duration. After the unitary operation, the logic-qubits will acquire the phase-shifts [32] j11iLL ! eig11 j11iLL , j10iLL ! eig10 j10iLL , j01iLL ! ig01 e j01iLL , and j00iLL ! eig00 j00iLL . As a result, the controlled-phase gate on logic-qubits can be written as 2 ig11 3 0 0 0 e 6 0 eig10 0 0 7 6 7 (11) U^ cp ¼ 6 7, 4 0 0 eig01 0 5 0

0

0

eig00

where g11 ¼ ðG  1 þ 2  3 þ 4 Þtcp =_, g10 ¼ ðG  1 þ 2 þ 3  4 Þtcp =_, g01 ¼ ðG þ 1  2  3 þ 4 Þtcp =_ and g00 ¼ ðG þ 1  2 þ 3  4 Þtcp =_.

ARTICLE IN PRESS Z.-B. Feng, X.-D. Zhang / Physica E 40 (2008) 878–882

The operation of switching on the inter-subsystem coupling G will also result in the changing of dcm simultaneously. We consider the following case to specify this effect. When all the gate voltages V gk are operated at the degeneracy point, from Eqs. (3) and (4), the charging energies will be 1 ¼ 4 ¼ 0, 2 ¼ 3 ¼ dcm , in which dcm varies only with V c P as dcm ¼ ðDm =2Þ cosða1 pV c =eþ a2 Þ cosð2pcÞ, where a1 ¼ k C mk ð1  cÞ and a2 ¼ 6cp are constants. With such a capacitive coupling between the boxes 2 and 3, the effects of dcm on boxes 1 and 4 can be ignored. IfR dcm is a time-dependent parameter, dynamic t phases i 0a dcm ðtÞ dt=_ [16] will be acquired only for the charge states j1ið2;3Þ and j0ið2;3ÞR, where ta is the duration. We t note for simplicity that fa ¼ 0 a dcm ðtÞ dt=_, and then logicqubits will acquire phases as j1iaL ! eifa j1iaL , j0iaL ! eifa j0iaL , j1ibL ! eifa j1ibL and j0ibL ! eifa j0ibL . After the switching operation, U^ cp will operate on logicqubit states with initial phases fj11iLL , e2ifa j10iLL , e2ifa j01iLL , j00iLL g. Since a universal quantum gate is irrelevant to the initial states, the switching time direction will not affect the logic-qubits gate operation. So far, based on the SCQs with controllable couplings, a universal set of gates in the encoded DFSs can be obtained. 3. Discussion and conclusion For SCQs that are operated in the charging regime, the fluctuations of the flux through the dc-SQUID loops couple to qubit variable s^ x , which is proved not to be the dominant noise [31]. The relatively strong decoherence effects are usually caused by the charge noises, including Ohmic noise originated from external control circuits and low-frequency 1=f noise coming from the background charge fluctuations [29–31]. Generally one can model the noises as oscillator baths. The Hamiltonian of the closed system can be characterized by spin-boson model H^ ¼ H^ S þ H^ B þ H^ SB , where H^ S and H^ B are the system and bath energies. Since each charge qubit is either coupled to an individual or a common (collective) noise [31], the system-bath interaction takes the form [15] X ^ H^ SB ¼ ðs^ zðiÞ þ s^ zðjÞ Þ  B^ c þ s^ ðkÞ z  Bk , k¼i;j

where B^ c and B^ k are correspondingly the common and independent bath operator. According to the encoded DFS, the effect of common bath will be eliminated. Therefore, for quantum gate operations, the enhancement of gate fidelity may be possible. In order to remove the collective charge noise, Zhou et al. [33] proposed a robust quantum information processing on pair-encoded charge qubits, by using two closely spaced cooper-pair boxes with a common bias lead. In two strongly capacitively coupled

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charge qubits, there exists common noise due to the correlation between charge fluctuations of the two boxes. It is believed helpful for significantly suppressing the decoherence effect [34]. If the four charge qubits with identical device parameters are subjected to the same environment as much as possible, the effect of the independent bath will be suppressed, even if there is a small symmetry breaking perturbation parameterized by Z in the order of OðZÞðZ51Þ [35]. In summary, we propose a theoretical scheme to realize QC in the DFS with SCQs. By taking advantage of controllable couplings, a universal set of gates can be constructed. In our scheme, the computational bases are all encoded in the subspace immune to the s^ z -type of decoherence. Taking into account the effects of charge noise in SCQs, we qualitatively analyze the value of DFS scheme combating certain decoherence effects. We hope this work is helpful to experimentally implement high fidelity quantum gate operations with DSJC. References [1] Y. Makhlin, G. Scho¨n, A. Shnirman, Rev. Mod. Phys. 73 (2001) 357. [2] Y. Nakamura, Y.A. Pashkin, J.S. Tsai, Nature (London) 398 (1999) 786; J.R. Friedman, et al., Nature (London) 406 (2000) 43. [3] C.H. van der Wal, et al., Science 290 (2000) 773; D. Vion, et al., Science 296 (2002) 886. [4] Y. Yu, et al., Science 296 (2002) 889. [5] Yu.A. Pashkin, et al., Nature (London) 421 (2003) 823; T. Yamamoto, et al., Nature (London) 425 (2003) 941. [6] A.J. Berklay, et al., Science 300 (2003) 1548; R. McDermott, et al., Science 307 (2005) 1299. [7] A. Izmalkov, et al., Phys. Rev. Lett. 93 (2004) 037003. [8] S.L. Zhu, Z.D. Wang, P. Zanardi, Phys. Rev. Lett. 94 (2005) 100502; S.L. Zhu, Z.D. Wang, K.Y. Yang, Phys. Rev. A 68 (2003) 034303; G.P. He, et al., Phys. Rev. A 68 (2003) 012315. [9] J.Q. You, J.S. Tsai, F. Nori, Phys. Rev. Lett. 89 (2002) 197902. [10] J. Lantz, et al., Phys. Rev. B 70 (2004) 140507(R). [11] M. Wallquist, et al., New J. Phys. 7 (2005) 178. [12] J.Q. You, J.S. Tsai, F. Nori, Phys. Rev. B 68 (2003) 024510. [13] F. Plastina, G. Falci, Phys. Rev. B 67 (2003) 224514. [14] J. Siewert, R. Fazio, Phys. Rev. Lett. 87 (2001) 257905. [15] M. Cholascinski, Phys. Rev. B 69 (2004) 134516. [16] D.V. Averin, C. Bruder, Phys. Rev. Lett. 91 (2003) 057003. [17] H. Paik, et al., IEEE Trans. Appl. Supercond. 15 (2005) 8223. [18] L.M. Duan, G.C. Guo, Phys. Rev. Lett. 79 (1997) 1953. [19] P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79 (1997) 3306. [20] D.A. Lidar, I.L. Chuang, K.B. Whaley, Phys. Rev. Lett. 81 (1998) 2594. [21] P. Zanardi, M. Rasetti, Phys. Lett. A 264 (1999) 94; J. Pachos, P. Zanardi, M. Rasetti, Phys. Rev. A 61 (2000) 010305. [22] L.M. Duan, J.I. Cirac, P. Zoller, Science 292 (2001) 1695. [23] S.L. Zhu, Z.D. Wang, Phys. Rev. Lett. 89 (2002) 097902; S.L. Zhu, Z.D. Wang, Phys. Rev. Lett. 91 (2003) 187902. [24] X.D. Zhang, et al., Phys. Rev. A 71 (2005) 014302. [25] L.-A. Wu, P. Zanardi, D.A. Lidar, Phys. Rev. Lett. 95 (2005) 130501. [26] X.D. Zhang, Q. Zhang, Z.D. Wang, Phys. Rev. A 74 (2006) 034302. [27] L. Viola, et al., Science 293 (2001) 2059. [28] E. Knill, R. Laflamme, L. Viola, Phys. Rev. Lett. 84 (2000) 2525; P. Zanardi, Phys. Rev. A 63 (2001) 012301; J. Kempe, et al., Phys. Rev. A 63 (2001) 042307.

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