Generation of quantum-dot cluster states with superconducting transmission line resonator
Zhi-Rong Lin, Guo-Ping Guo, Tao Tu, Fei-Yun Zhu, Guang-Can Guo
aa r X i v : . [ qu a n t - ph ] J a n Generation of quantum-dot cluster states with superconductingtransmission line resonator
Zhi-Rong Lin, Guo-Ping Guo, ∗ Tao Tu, † Fei-Yun Zhu, and Guang-Can Guo
Key Laboratory of Quantum Information,University of Science and Technology of China,Chinese Academy of Sciences, Hefei 230026, People’s Republic of China (Dated: October 30, 2018)
Abstract
We propose an efficient method to generate cluster states in spatially separated double quantumdots with a superconducting transmission line resonator (TLR). When the detuning between thedouble-dot qubits transition frequency and the frequency of the full wave mode in the TLR satisfiessome conditions, an Ising-like operator between arbitrary two separated qubits can be achieved.Even including the main noise sources, it’s shown that the high fidelity cluster states could begenerated in this solid system in just one step.
PACS numbers: 03.67.Lx, 42.50.Pq, 42.50.Dv ∗ Electronic address: [email protected] † Electronic address: [email protected] ntroduction.— Quantum entanglement is the root in quantum computation [1], quantumteleportation [2], quantum dense coding [3], and quantum cryptography [4]. However, it’schallenging to create multi-particle entangled states in experiment. In 2001 Briegel andRaussendorf introduced a highly entangled states, the cluster states [5], which can be used toperform universal one way quantum computation. Up to now, various schemes are proposedto generate cluster states in many different types of physical systems. Especially, it hasbeen argued that the cluster states can be generated effectively in solid state system, suchas superconductor charge qubit [6, 7, 8] and semiconductor quantum dot [9, 10, 11].Electron spins in semiconductor quantum dots are one of the most promising candidatesfor a quantum bit, due to their potential of long coherence time [12, 13, 14]. Producingcluster states in quantum dots, has been discussed within Heisenberg interaction model [9]and Ising-like interaction model [10], where the long-term interaction inversely ratios to thedistance between non-neighboring qubits. Recently Childress and Taylor et al. introduceda technique to electrically couple electron charge states or spin states associated with semi-conductor double quantum dots to a TLR via capacitor [15, 16]. The qubit is encoded onthe quantum double-dot triplet and singlet states. The interaction Hamiltonian between thequbits and the TLR is a standard Jaynes-Cumming (JC) model [17]. A switchable long-range interaction can be achieved between any two spatially separated qubits with the TLRcavity field. This technique open a new avenue for quantum information implementation.In this work, we find when the detuning between the qubits transition frequency and thefrequency of the full wave mode in the TLR satisfies some conditions, an Ising-like operatorbetween arbitrary two separated qubits can be achieved from the JC interaction. The highlyentangled cluster states can be generated one step with the auxiliary of an oscillating electricfield. Finally, we discuss the feasibility and the cluster states fidelity of our scheme.
Preparation of cluster states.—
The system we study includes N identical double-dotqubits capacitively coupling with a TLR by C c , as shown in Fig. (1). The TLR is coupledto input/output wiring with a capacitor C e to transmit signals. Two electrons are localizedin double-dot quantum molecule. The two electrons charge states, spin states and thecorresponding eigenenergies are controlled by several electrostatic gates.An external magnetic field is applied along axis z . At a large static magnetic field (100mT), the spin aligned states ( | (1 , T + i = |↑↑i , | (1 , T − i = |↓↓i ) are splitted from spinanti-aligned states ( | (1 , T i = ( |↑↓i + |↓↑i ) / √ | (1 , S i = ( |↑↓i − |↓↑i ) / √
2) due to2 xt B c C D e Cinput/output wire Double-dotsTLR
FIG. 1: (Color online) Schematic diagram of a TLR and several double quantum dots coupledsystem. A detailed circuit representation of the TLR cavity (blue) can be found in Fig.1 of Ref.[18].The double dots are biased with an external potential ∆, and capacitive coupling C c with the TLR.The TLR is connected to the input/output wiring with a capacitor C e . Zeeman splitting. The notation ( n l , n r ) indicates n l electrons on the ”left” dot and n r electrons on the ”right” dot. In addition to the (1 ,
1) subspace, the doubly-occupied state | (0 , S i is coupled to | (1 , S i via tunneling T c . | (0 , S i and | (1 , S i have a potentialenergy difference of ∆. Due to the tunneling between the two adjacent dots, the | (1 , S i and | (0 , S i hybridize. We can get the double dots eigenstates: | + i = − sin θ | (1 , S i + cos θ | (0 , S i , (1) |−i = cos θ | (1 , S i + sin θ | (0 , S i , (2)where tan θ = − T c / (Ω + ∆) and Ω = p T c + ∆ . Ω is the energy gap between theeigenstates | + i and |−i . We can choose ∆ = 0 in order to suppress the fluctuations incontrol electronics, then | + i = ( | (1 , S i + | (0 , S i ) / √ |−i = ( | (1 , S i − | (0 , S i ) / √ | + i and |−i .An oscillating electric field, which frequency is coincidental with the qubit energy gap,is applied to the left side gate of the double quantum dots. The single-qubit operation3amiltonian in the interaction picture can be expressed as H i = η | (1 , S i h (1 , S | = η ( I + σ xi ) = ησ xi , (3)where η = ˆ d · ˆ E , ˆ d is the dipole of left quantum dot and ˆ E is the oscillating electric field.We consider a TLR of length L, with capacitance per unit length C , and characteristicimpedance Z . Neglecting the higher energy modes, we can only consider the full wave mode,with the wavevector k = πL , and frequency ω = kC Z [16]. The TLR can be described by theHamiltonian H cavity = ωa † a , where ~ = 1, a † , a are the creation and annihilation operatorsfor the full wave mode of the TLR. In the interaction picture, the interaction between Nqubits and the TLR can be described by the Hamiltonian [17] H int = g N X i =1 ( e iδt a † σ − i + e − iδt aσ + i ) , (4)where σ + i = | + i h−| , σ − i = |−i h + | , σ xi = σ + i + σ − i , δ = ω − Ω. Here we can presently assumethe coupling strength is homogenous. The overall coupling coefficient can be described by[16] g = ω C c C tot s z R Q , (5)where R Q = h/e ≈
26 kΩ. C tot is the total capacitance of the double-dot.If the interaction time τ satisfies δτ = 2 kπ, (6)the evolution operator for the interaction Hamiltonian (4) can be expressed as [19] U ( τ ) = exp ( − i λ τ ( N X i =1 σ xi ) ) = exp ( − iλτ N X j>i =1 σ xi σ xj ) , (7)where λ = g / δ .When ∆ is changed to zero, the coupling coefficient between the qubits and TLR ismaximal. An oscillating electric field is applied to all the qubits at the same time. In theinteraction picture, the total Hamiltonian of the system can be written as H tot = g N X i =1 ( e iδt a † σ − i + e − iδt aσ + i ) + η N X i =1 σ xi . (8)4hen the operation time τ satisfies the condition (6), we can obtain the total evolutionoperator U ( τ ) = exp ( − iητ N X i =1 σ xi − iλτ N X j>i =1 σ xi σ xj ) . (9)When η = ( N − λ , the total evolution operator is given by U ( τ ) = exp ( − iλτ N X j>i =1 σ xi σ xj . (10)In order to generate the cluster states, the initial state of N qubits should be prepared inthe state N Ni =1 |−i i = √ N Ni =1 ( | i i + | i i ), where | i i = √ ( |−i i ±| + i i ) are the eigenstatesof σ xi with the eigenvalues ±
1. Next we would discuss how to prepare the initial state inexperiment. Firstly we can prepare the two electrons in double quantum dots to the state | (0 , S i at a large positive potential energy difference ∆ [20]. Then the | (0 , S i can bechanged to the state |−i = ( | (1 , S i − | (0 , S i ) / √ λτ = (2 n + 1) π, (11)with n being integer, the spatially separated double quantum dots can be generated to thecluster states | Ψ i N = 12 N/ N O i =1 ( | i i ( − N − i N Y j = i +1 σ xi + | i i ) . (12)The effective coupling coefficient g T c Ω can be tuned by external potential ∆. When ∆ ischanged, the states | i would change according to Eq. (1), (2), but the expression (12)of the cluster states is unchanged. When the cluster states is generated at the time of τ ,we can remove the oscillating electric field and change ∆ to discouple all the qubits to theTLR. Then the cluster states can be preserved. Feasibility of the scheme.—
The sample of the TLR and quantum dots coupled system canbe obtained in a two-step fabrication process on a GaAs/AlGaAs heterostructure. Firstly,quantum dots are formed in the two-dimensional electron gas below the surface, using elec-tron beam lithography and Cr-Au metallization. Then the TLR is fabricated by conventionaloptical lithography [21]. The main technical challenges for experimental implementation of5ur proposal are the design and nanofabrication of the sample [22]. The diameter of thequantum dot is about 400 nm, and the corresponding capacitance of the double-dot C tot isabout 200 aF. The distance between the two double-dot molecules should be 4 µ m which istenfold of the distance between two quantum dots within a double-dot. Thus we can neglectthe interaction between the double-dot molecules safely [23]. Since the energy gap between | + i and |−i is about 10 µ eV at the operation point, the experimental manipulation shouldbe implemented in dilution refrigerator with temperature below 100 mK. Both the condi-tions (6) and (11) are satisfied whenever δ = g p k/ (2 n + 1). From Eq. (5) the couplingcoefficient g can be up to ω/
16 with a large coupling capacitor C c ≈ C tot ≈
400 aF. For k = 1, n = 0, ω/ π = 2 GHz and g / π = 125 MHz, the operation time of the generationof cluster states is τ = πδ = 4 ns. Decoherence.—
In our system, the main decoherence processes are the dissipation of theTLR, the spin dephasing, charge relaxation and additional dephasing of the double-dotmolecules. The dissipation of the TLR occurred through coupling to the external leadscan be described by the photon decay rate κ = ω/Q , where Q is the quality factor. For Q = 1 × , ω = 2 π × /κ ≈ µ s is 4orders longer than the operation time τ . Therefore the cavity loss can be neglected in oursituation.Nuclear spins are one of the main noise sources in semiconductor quantum dots viahyperfine interaction. The hyperfine field can be treated as a static quantity, because theevolution of the random hyperfine field is several orders slower ( > µ s) than the electronspin dephasing. In the operating point, the most important decoherence due to hyperfinefield is the spin dephasing between the states | (1 , S i and | (1 , T i . By suppressing nuclearspin fluctuations [26], the spin dephasing time obtained by quasi-static approximation can be T ∗ = ~ / ( gµ B h ∆ B zn i rms ) ≈ µ s, where ∆ B zn is the nuclear hyperfine gradient field betweentwo dots and rms denotes a root-mean-square time-ensemble average. The coupling to thephonon bath will lead to the relaxation of the charge freedom. Using the spin-boson model,the relaxation time of the qubits can be obtained by Fermi-Golden rule [15]. The chargerelaxation time T is about 1 µ s at the operation point. Additional dephasing is assumed toarise from the low frequency fluctuations of the control electronics, which typically have the1 /f spectrum. In our system, it’s assumed that the origin of 1 /f noise is the random driftof the gate bias when an electron tunnel in or out of the metallic electrode. Assuming 1 /f T ,α ∼ Ω T ,bare isabout 100 ns at the optimal working point (∆ ≈ T ,bare will be discussed below indetail and can be up to 10 ns. Thus the total operation time of the present proposal τ ≈ Fidelity of the cluster states.—
For simplicity, we assume the control electronics fluc-tuations are Gaussian. These noises would lead to the fluctuations of the parameter λ via the electric potential difference ∆. Suppose ∆ i ( t ) = ∆ + ǫ i ( t ), h ǫ i ( t ) i = 0, h ǫ i ( t ) ǫ j ( t ′ ) i = R S ij ( ω ) e iω ( t − t ′ ) dω ( i labeling the i -th qubit). The fluctuations of the os-cillating electric field would result in the fluctuations of the parameter η . The fluctuationsof λ and η would add an unwanted phase θ i to the desired value π [27]. Including thefluctuations, the evolution operator (10) should be rewritten in the form of U ( τ ) = exp ( − iπ N X j>i =1 σ xi σ xj exp ( − i Z τ δη ( t ) dt N X i =1 σ xi ) exp ( − i Z τ δλ ( t ) dt N X j>i =1 σ xi σ xj ) , (13)where λ ( t ) = λ + δλ ( t ) and η ( t ) = η + δη ( t ). Since δλ + δηN − , δη + ( N − δλ are largerthan δλ , δη , we can write the unwanted phase θ i = R τ δλ ( t ) + δη ( t ) N − ) dt = θ ,i + θ ,i , where θ ,i = R τ δλ ( t ) dt and θ ,i = R τ δη ( t ) N − dt .Since δ , λ satisfy Gaussian distribution, θ ,i , θ ,i , θ i have Gaussian distribution G (0 , σ ,i ), G (0 , σ ,i ), G (0 , σ i ). Ignoring the correlative fluctuations, the variance of θ ,i at the optimalworking point is σ ,i = ( 2 g Ω δ ) (cid:28) ( Z τ ǫ ( t ) dt ) (cid:29) , (14)where (cid:10) ( R τ ǫ ( t ) dt ) (cid:11) = ( R S i ( ω ) dω ) τ +2( R S i ( ω ) sin ωτωτ dω ) τ . (15)For the low frequency noise, S i ( ω ) has a high frequency cutoff γ ≪ τ . Therefore wecan get (cid:10) ( R τ ǫ ( t ) dt ) (cid:11) = 3( R S i ( ω ) dω ) τ . Assuming T ,bare = R S i ( ω ) dω , we can obtain thevariance σ ,i = g δ ( τ Ω T ,bare ) . Taking T ,bare ≈
10 ns from the Ref. [20, 24, 25], the varianceof θ ,i is σ ,i = 0 . π . The fluctuations of the oscillating electric field root in the electronicsnoise. The fluctuations can be reduced in a small value with better high- and low-frequency7
10 15 20 25 300.960.970.980.991
N qubits F i d e lit y FIG. 2: The fidelity of N-qubit cluster states. filtering technique. Supposing σ ∆ η / ∆ η ≈ θ ,i is σ ,i = 0 . π . So θ i hasan Gaussian distribution G (0 , (0 . π ) ). The fidelity of N qubits cluster states is calculatedaccording to the formula F = | − N P z i Q N − j =1 ( R √ πσ i e − θ j σ i e iθ j dθ j ) z j z j +1 | from Ref. [27], asshown in Fig. (2). The fidelity of a 30-qubit cluster states can be 96 . Conclusion and Discussions.—
Distinguished from cavity quantum electrodynamics inatomic quantum information processing, our scheme can realize the long-range interactionamong the double-dot molecules with the TLR in a solid micro-chip device. This techniquecan couple the static qubit in the solid state system to the flying qubit (the cavity photon)[15]. Compared with other schemes, the present proposal based on quantum dot moleculeshas four potential advantages: integration and scaling in a chip, easy-addressing, high con-trollability, and long coherence time associated with the electron spin. As discussed above,the preparation of the initial state can be easily implemented without inter-qubit couplingin our scheme. When the initial state has been prepared, the quantum-dot cluster statescan be produced with only one step. The cluster states can be preserved easily by switchingoff the coupling between the qubits and TLR cavity field.In conclusion, we proposed a realizable scheme to generate cluster states only one step ina new scalable solid state system, where the spatially separated semiconductor double-dotmolecules are capacitive coupling with a TLR. An effective, switchable long-range interactioncan be achieved between any two double-dot qubits with the assistance of TLR cavity field.The experimental related parameters and the possible fidelity of generated cluster stateshave been analyzed. Due to the long relaxation and dephasing time at the optimal working8oint, the present scheme seems implementable within today techniques.This work was funded by National Basic Research Programme of China (Grants No.2009CB929600, No. 2006CB921900), the Innovation funds from Chinese Academy ofSciences, and National Natural Science Foundation of China (Grants No. 10604052,No.10804104, No. 10874163). [1] J. Gruska, Quantum Computing (London: McGraw-Hill, New York, 1999).[2] C. H. Bennett et al. , Phys. Rev. Lett. , 1895 (1993).[3] C. H. Bennett, S. J. Wiesner, Phys. Rev. Lett. , 2881 (1992).[4] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[5] H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. , 910 (2001).[6] T. Tanamoto, Y. X. Liu, S. Fujita, X. Hu and F. Nori, Phys. Rev. Lett. , 230501 (2006).[7] J. Q. You, X. B. Wang, T. Tanamoto and F. Nori, Phys. Rev. A. , 052319 (2007).[8] Z. Y. Xue and Z. D. Wang, Phys. Rev. A. , 064303 (2007).[9] M. Borhani and D. Loss, Phys. Rev. A. , 034308 (2005).[10] G. P. Guo, H. Zhang, T. Tu and G. C. Guo, Phys. Rev. A. , 050301(R) (2007).[11] Y. S. Weinstein, C. S. Hellberg, and J. Levy, Phys. Rev. A. , 020304(R) (2005).[12] J. M. Taylor et al. , Phys. Rev. B , 035315 (2007).[13] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. , 016601 (2004).[14] A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B , 12639 (2000).[15] J. M. Taylor and M. D. Lukin, arXiv: cond-mat/0605144.[16] L. Childress, A. S. Sørensen, and M. D. Lukin, Phys. Rev. A , 042302 (2004).[17] G. P. Guo, H. Zhang, Y. Hu, T. Tu, and G. C. Guo, Phys. Rev. A , 020302(R) (2008).[18] A. Blais et al. , Phys. Rev. A. , 032329 (2007).[19] S. B. Zheng, Phys. Rev. A. , 060303(R) (2002).[20] J. R. Petta et al. , Science , 2180 (2005).[21] A. Wallraff et al. , Nature , 162 (2004).[22] D. I. Schuster, PhD Thesis, Yale University (2007).[23] G. P. Guo et al. Eur.Phys. J. B , 141 (2008).[24] A. S. Bracker et al. , Phys. Rev. Lett. , 047402 (2005).
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