Abstract We propose a new and feasible scheme to implement quantum gates in decoher-ence-free subspaces (DFSs) with Josephson charge qubits situated in a circuit QED architec-ture. Based on the resonator-assisted interaction, the controllable interqubit couplings occuronly by tuning the individual flux biases, by which we obtain the DFS-encoded universalquantum gates. Compared with the non-DFS situation, we numerically consider the robust-ness of the DFS-encoded scheme that can be insensitive to the collective noises. Thus theprotocol may perform the fault-tolerant quantum computing with Josephson charge qubits.
Due to flexible manipulations and potential scalability, Josephson qubits have achieved theremarkable progress in quantum information processing [1, 2]. As one of the basic elements,charge qubit can be conveniently addressed by the external parameters such as gate voltagesand magnetic fluxes, and then has attracted considerable attention in the past decade [3–5].Recently, circuit quantum electrodynamics (QED) emerged in superconducting nanocircuitsgreatly promoted the development of quantum state engineering [6–8]. Taking advantage ofthe transmission line resonator (TLR) in the circuit QED, the strong couplings between thequantized electromagnetic field and the charge qubits have been realized experimentally atmicrowave frequency range [9, 10].
Despite the achieved progress, however, decoherences caused by the environmentalnoises impair greatly the capability of the qubits to process quantum information [11–13].How to construct the fault-tolerant schemes is still a crucial issue for the practical quan-tum computers. Many robust strategies to resist noise effects have been put forward, such as
Z.-B. Feng (�) · R.-Y. Yan · C. Zhang · L. FanElectric and Information Engineering College, Xuchang University, Xuchang 461000, Chinae-mail: [email protected]
Fig. 1 Schematic diagram of the charge qubits in a circuit QED device
geometric quantum computation [14, 15], topologically protected qubits [16, 17], etc. In par-ticular, encoding qubits into decoherence-free subspace (DFS) is an interesting way to avoidquantum errors [18–21]. Towards implementing the fault-tolerant schemes, the controllableinterqubit couplings and the optimal quantum operations are two important aspects. The for-mer one is determinant in scaling up qubit number, and the latter one affects the gate fidelity.Thus an effective combination of the above two aspects is vital to realize the fault-tolerantquantum operations on multiqubit systems.
To obtain the robust quantum computing in an optimal way, in this paper we propose atheoretical scheme to perform the quantum gates in DFSs with charge qubits. Many qubitsare arrayed in a circuit QED acting as quantum data bus. The controllable interqubit cou-plings between any pair of selected qubits can be achieved by adjusting the static flux biasesonly. Based on the switchable couplings, we implement the universal gate operations onthe DFS-encoded logic qubits to remove the collective noises. Moreover, we numericallyanalyze the robustness of the DFS-encoded quantum gates. Taking into account the opti-mal operation, the scheme may provide the possibility to obtain the fault-tolerant quantumcomputing with Josephson charge qubits.
The paper is organized as follows. In Sect. 2, we present charge qubits in a circuit QEDdevice. In Sect. 3, the resonator-assisted interqubit coupling is achieved controllably. TheDFS-encoded logic gates are given in Sect. 4. We numerically consider the robustness of theDFS-encoded quantum operations in Sect. 5. Finally, discussions and conclusions are drawnin Sect. 6.
2 Charge Qubits in a Circuit QED
A high Q one-dimensional TLR of length Lx along x direction, as schematically shown inFig. 1, has a single-mode frequency ωr and generates the quantized standing-wave field [22,23]. The TLR is connected to the input (output) port of the waveguide through a capacitorCi (Co). Many Josephson charge qubits are situated at the antinodes and capacitively cou-pled to the TLR [24]. We assume that the capacitances are identical for simplicity. Then thepresent architecture forms the called circuit QED system. In each charge qubit, a Cooper-pair box includes excess Cooper-pairs n, and is connected to a segment of a superconduct-ing loop via two symmetric Josephson junctions (with coupling energy EJ0 and capacitanceCJ ). The split-pair geometry is designed to adjust the effective Josephson energy EJk by anexternal flux �k threading the SQUID loop, k = 1,2, . . .N .
2284 Int J Theor Phys (2012) 51:2282–2290
A static gate voltage Vk applied to the capacitance C induces offset charge numbernk = CVk/2e. The Hamiltonian describing the static system reads H(k)
c = Ec(n − nk)2 −
EJk cos θ . The characteristic charge energy is Ec = 2e2/C� , with C� = (C + 2CJ ) be-ing the total capacitance connected to the box. The effective Josephson coupling is EJk =2EJ0 cos(π�k/�0), in which �0 = h/2e denotes the flux quantum. And θ indicates the av-erage phase difference across the Josephson junctions. In the charging regime, the systemparameters satisfy Ec � EJ0, which makes only charge-number states |0〉k and |1〉k goodquantum numbers (near nk = 1/2). Within the basis {|0〉, |1〉}k , the Hamiltonian in spin-1/2representation is written as
H(k)c = Ec(1 − 2nk)σ̄kz/2 − EJkσ̄kx/2, (1)
where Pauli matrices are σ̄kz = |1〉k〈1| − |0〉k〈0| and σ̄kx = |1〉k〈0| + |0〉k〈1|.Using the energy eigenbasis {|e〉, |g〉}k of H(k)
c , we have H(k)c = �ωkσkz/2. The transition
frequency between |e〉k and |g〉k is ωk =√
[Ec(1 − 2nk)]2 + E2Jk/�, which is tunable via
the applied flux �k . The Pauli matrix takes the form σkz = |e〉k〈e|− |g〉k〈g|, where thelevel states are |e〉k = − sin θk
2 |0〉k + cos θk
2 |1〉k and |g〉k = cos θk
2 |0〉k + sin θk
2 |1〉k , with θk =tan−1[EJk/Ec(1 − 2nk)]. At the charge degeneracy point nk = 1/2, the chosen qubit statesbecome
|e〉k = (|1〉k − |0〉k)/√
2,
|g〉k = (|1〉k + |0〉k)/√
2.(2)
Besides the dc voltage Vk , a quantum part Vq generated by the TLR is Vq =√
�ωr
Lxc(a† +a)
at the antinodes [6], where c is the transmission line capacitance per unit length, a†(a) isthe creation (annihilation) operator associated with the resonator. The quantized voltage Vq
is coupled to the kth Cooper-pair box through the capacitance C, and its interaction Hamil-
tonian is given by H(k)cr = −λ(a† + a)σ̄kz, with λ = eC
C�
√�ωr
Lxcbeing coupling coefficient.
Choosing the qubit basis of (2) yields H(k)cr = λ(a† + a)(σ−
k + σ+k ), in which the inver-
sion operators are defined as σ+k = |e〉k〈g| and σ−
k = |g〉k〈e|. Adopting the rotating-waveapproximation, we neglect the fast oscillation terms and then obtain
H(k)cr = λ
(a†σ−
k + aσ+k
), (3)
which has the usual Jaynes-Cummings form.
3 Controllable Interqubit Coupling
Considering two arbitrary charge qubits, say k = 1 and 2. In the dispersive conditions, thedetunings are much larger than coupling strengths, δ1,2(= ωr − ω1,2) � λ. In this case, thecombined system is described by the total Hamiltonian Ht = H0 + H1 [25], where
H0 = ωra†a + ω1σ1z/2 + ω2σ2z/2,
H1 = 12σ1z/2 + λ∑k=1,2
(e−iδk t a†σ−
k + h.c.), (4)
here we address a detuning 12 = ω1 − ω2 and take � = 1. Given the cavity field is initiallyin the vacuum state, the effective Hamiltonian of H1 in the interaction picture is [26]
Heff = 12σ1z/2 + J
[ ∑k=1,2
|e〉k〈e| +(σ+
1 σ−2 + σ−
1 σ+2
)], (5)
Int J Theor Phys (2012) 51:2282–2290 2285
where the transverse coupling is J = λ2
2 ( 1δ1
+ 1δ2
). As a result, the coupling between twoqubits is achieved by the virtual exchange of photons with cavity bus.
In the two-qubit basis {|gg〉, |eg〉, |ge〉, |ee〉}, the effective Hamiltonian of (5) takes thematrix form
Heff =
⎛⎜⎜⎜⎝
−122 0 0 0
0 122 + J J 0
0 J −122 + J 0
0 0 0 122 + 2J
⎞⎟⎟⎟⎠ . (6)
The symbols in each product state indicate the quantum states associated with qubits 1 and2, respectively. For an arbitrary state vector ψ(t) = Cgg|gg〉+Ceg|eg〉+Cge|ge〉+Cee|ee〉,we can get its time evolution which is governed by the differential equations
id
dtCgg = −12
2Cgg,
id
dtCeg =
(12
2+ J
)Ceg + JCge,
id
dtCge = JCeg +
(J − 12
2
)Cge,
id
dtCee =
(12
2+ 2J
)Cee,
(7)
where Cgg , Ceg , Cge and Cee are the superposition coefficients meeting the normalizationcondition.
By numerically solving (7), we consider the coherent evolutions of quantum states. Sup-pose that ω1 acts as the stationary transition frequency, and ω2 is tunable by means of �2.The change of detuning 12 is dependent on ω2 only. Applying the experimentally availableparameters [7, 27, 28], ω1/2π = 8.5 GHz, ωr/2π = 10 GHz and λ/2π = 105 MHz, we havethe detunings δ1 and δ2 much larger than the cavity-induced coupling coefficient λ. There-fore the needed dispersive conditions δ1,2 � λ are satisfied well. As numerically shown inFig. 2(a)–(d), we give the coherent population evolutions of |eg〉 and |ge〉 in the resonantand nonresonant cases, respectively, with the systems being initially in state |eg〉. Obvi-ously, the probability Pge(= |Cge|2) transferred from Peg(= |Ceg|2) will be reduced greatlywith increasing the detuning 12.
To further show the influence of detuning 12 on the state evolutions, we numerically plotthe occupied probabilities Peg(tn) and Pge(tn) as functions of 12, see Fig. 3. Here we treata specific time tn(= π/2J ) and the system state is initially in |eg〉 as well. When the qubitlevels are in resonance, namely 12 = 0, there have Pge(tn) = 1 and Peg(tn) = 0, which areconsistent with the results of Fig. 2(a). The coherent probability transfer between |eg〉 and|ge〉 will be reduced significantly with increasing the detuning 12. When the qubit transi-tion frequencies are 0.5 GHz apart, 12/2π = 0.5 GHz, the probability Pge transferred fromPeg will be lower than 1.4 × 10−3, displayed in the subfigure. At the large detuning regime12 � J , energy conservation suppresses the flip-flop interaction, and then the interqubitstate transfer is turned off effectively [7].
Thereby, only by adjusting the qubit transition frequencies in resonance or large detun-ing, we achieve a controllable interqubit coupling effectively, which is highly desirable tomanipulate many qubits.
2286 Int J Theor Phys (2012) 51:2282–2290
Fig. 2 Coherent evolutions of quantum states |eg〉 and |ge〉, with 12 = 0 in (a), 12 = 20 MHz in (b),12 = 50 MHz in (c), 12 = 100 MHz in (d)
Fig. 3 The occupied populationsas functions of the detuning 12for the given time tn
4 DFS-Encoded Quantum Gate Operations
The two qubits 1 and 2 span a Hilbert space as {|gg〉, |ge〉, |eg〉, |ee〉}, from which we choosea subspace {|ge〉, |eg〉}. Logic qubit states are encoded as |0〉L = |ge〉 and |1〉L = |eg〉, whichconstitute the computational basis. Based on the symmetry of system-bath interaction, thesubspace referred to as the DFS can be immune to certain collective noises [29, 30]. Theconsidered DFS {|ge〉, |eg〉} is insensitive to the σz-type collective noises. The reason for that
Int J Theor Phys (2012) 51:2282–2290 2287
is as follows. When the system-bath interaction has the form Z⊗ B̂c , any state |ψ〉 within thebasis {|0〉L, |1〉L} always has (Z ⊗ B̂c)|ψ〉 = 0 due to Z|ψ〉 = 0, where Z = σ1z + σ2z, andB̂c is a random operator associated with the collective bath. Thus the effects of the σz-typecollective bath on the logic qubits |0〉L and |1〉L can be removed symmetrically.
Hereafter, using the controllable interqubit coupling, we perform the universal gate op-erations on the DFS-encoded logical qubits. Firstly, we construct single-logic-qubit gates.By tuning the external flux �2 applied to the charge qubit 2, we get the resonant transitionfrequencies ω2 = ω1, and then the detuning 12 = 0. In this case, the effective Hamiltonianof (5) will be reduced as
Heff = J
[ ∑k=1,2
|e〉k〈e| +(σ+
1 σ−2 + σ−
1 σ+2
)], (8)
here J = λ2/δ1 because of δ2 = δ1. In the logic qubit basis {|1〉L, |0〉L}, the evolution opera-tor UL = e−iHeff t with respect to Heff is represented as
UL = e−iJ t
(cos(J t) −i sin(J t)
−i sin(J t) cos(J t)
). (9)
Here the identical phase factors e−iJ t are acquired by |1〉L and |0〉L. Obviously, two non-commutable single-logic-qubit gates can be accomplished by appropriately controlling theevolution time [20]. In particular, a not-gate operation can be realized after a duration time tn.
Next, we perform a nontrivial two-logic-qubit controlled phase gate. Choosing other twocharge qubits 3 and 4, similarly, we encode logic-qubit states as |0〉L = |ge〉 and |1〉L = |eg〉.Then the DFS-encoded two-logic-qubits are |00〉LL = |gege〉, |01〉LL = |geeg〉, |10〉LL =|egge〉 and |11〉LL = |egeg〉. By virtue of an auxiliary state |g〉5 of qubit 5, we implementa controlled-phase gate operation on the selected two-logic-qubits [20, 31]. Only by tuningthe external fluxes �5 and �3 that are applied to the charge qubits 5 and 3, respectively, wehave the resonant transition frequencies ω5 = ω3 = ω1. Therefore the effective interactionHamiltonian describing the three-qubit interaction is
Heff = J
( ∑k=1,3,5
|e〉k〈e| +k �=p∑
k,p=1,3,5
σ−k σ+
p
). (10)
In the light of Hamiltonian (10), the quantum state evolutions will be |00〉LL|g〉5 →|00〉LL|g〉5, |01〉LL|g〉5 → |01〉LL|g〉5, |10〉LL|g〉5 → |10〉LL|g〉5 and |11〉LL|g〉5 →eiϕ11 |11〉LL|g〉5, as long as the duration time satisfies tcp = 2π/3J . And the controlled-phase factor is ϕ11 = −2π/3. Obviously, the auxiliary qubit 5 becomes decoupled fromother qubits, and thus the controlled-phase operation on the logic qubits is obtained as
ULL =
⎛⎜⎜⎜⎝
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 eiϕ11
⎞⎟⎟⎟⎠ . (11)
Consequently, taking advantages of the controllable interqubit couplings, we haveachieved a set of universal quantum gates using the DFS-encoded logic qubits, which con-tains two noncommutative single-logic-qubit gates and a two-logic-qubit controlled-phasegate.
2288 Int J Theor Phys (2012) 51:2282–2290
5 Robustness of the DFS-Encoded Gates
In the followings, we analyze the improved robustness by means of the DFS-encoded ap-proach. In general, environmental noises acting on qubits can be classified into two types.One is individual noise, another is collective noise. Both types of noises can be coupled toeach qubit k, whose interaction Hamiltonian has the form [19, 31]
Hsb =∑
k
σkz ⊗ B̂k +∑
k
σkz ⊗ B̂c, (12)
where B̂k and B̂c denote the individual and collective noises, respectively. Since the DFS-encoded logic qubits are immune to the collective noises, the B̂c induced decoherence effectscan be removed naturally. Thus the total decoherence effects may be depressed remarkably,which is helpful to achieve the robust gates. Physically, the collective noises may be origi-nated from the fluctuations of the external control variables and the intrinsic system param-eter. For the former, it contains low-frequency voltage fluctuations coupling identically toqubit variables σ̄kz. Additionally, the fluctuations of the fluxes threading the loops are cou-pled to σ̄kx . Since the Cooper-pair boxes are operated in charge regime, such effects seemnot to be dominant. For the latter, the identical background charge and current fluctuationsmay behave as the collective noise sources.
Quantitatively, the decoherence effects can be simulated by the gate fidelity, which isdefined as F = √〈ψi |ρ|ψi〉 [32], where |ψi〉 is ideal output state without the decoherenceeffects, ρ = |ψ〉〈ψ | represents density matrix with respect to the realistic state |ψ〉. Now,using the standard quantum theory of damping, we consider the effects of decoherence pro-cesses on the single-logic-qubit gate operations. In the DFS-encoded case, after tracing outthe bath degrees of freedom, we obtain the reduced density matrix ρd associated with thetwo-qubit, whose dynamical evolution follows the Lindblad master equation [6],
ρ̇d = −i[Heff , ρd ] +∑k=1,2
γkD[σ−
k
]ρd +
∑k=1,2
γkφ
2D[σkz]ρd, (13)
where Heff has the form of (8), γk and γkφ are the respective relaxation and dephasingrates of individual qubits, D[L]ρd = (2LρdL
† − L†Lρd − ρdL†L)/2, with L = σ−
k and σkz.However, in the non-DFS case, the two qubits are affected by the collective noises besidesthe individual noises. Then the evolution of the reduced density matrix ρnd is governed by
ρ̇nd = −i[Heff , ρnd ] +∑k=1,2
γkD[σ−
k
]ρnd +
∑k=1,2
γkφ + γφ
2D[σkz]ρnd, (14)
where γφ is the common dephasing rate for both qubit 1 and 2. Due to the virtual exchangephoton, here we have neglected the effects of the photon leakages on the decoherence pro-cesses.
Compared to the non-DFS case, we restrict our attention to the robustness of the DFS-encoded gates by fighting against the decoherence effects caused by the σz-type collectivenoises. The fidelities in the DFS-encoded and non-DFS cases are denoted as Fd and Fnd ,respectively. Together with (8), the fidelities of the quantum operations in both cases can beobtained numerically by solving (13) and (14). As shown in Fig. 4, we numerically simulatethe fidelities with the same evolution time tn, and plot Fd and Fnd as functions of the collec-tive dephasing rate γφ . Here the relaxation and dephasing rates at the optimal point are cho-sen as γ1 = 0.02 MHz, γ2 = 0.05 MHz, γ1φ/2π = 0.1 MHz and γ2φ/2π = 0.2 MHz, respec-tively, which are accessible in current experiments [7]. Since the DFS-encoded qubits canbe immune to the collective noises, the fidelity Fd is independent of γcφ . Different from Fd ,
Int J Theor Phys (2012) 51:2282–2290 2289
Fig. 4 In the DFS-encoded andnon-DFS cases, dependences ofthe not-gate fidelities Fd and Fnd
on the collective dephasingrate γφ
the fidelity Fnd reduces greatly with increasing the parameter γφ . When γφ/2π = 0.5 MHz,the difference F(= Fd − Fnd) will be 2.36%. Further, we can estimate F for the two-logic-qubit controlled-phase gate. Due to the dissipative processes, F will be larger than4.72% after a longer gate time tcp . Comparatively, the DFS encoded gate operations can berobustly realized via eliminating the collective noises.
6 Discussions and Conclusions
The present protocol may have the following advantages. (i) The considered circuits can beflexibly operated only via the external fluxes, and then we can selectively and controllablyperform the logic gates on the encoded qubits, which is highly desirable to implement thescalable quantum information processing. (ii) In the previous DFS-encoded schemes [20,31], the level transitions were induced by ac voltage pulses with certain microwave fre-quencies. Actually, the ac gate pulses cause the unwanted fluctuations of the system levelsunavoidably, which are detrimental to quantum coherent operations. Compared with the pre-vious schemes, we utilize the static magnetic flux biases instead of ac gate pulses to obtainthe optimal operations. (iii) Through the exchange of virtual rather than real photons, theinterqubit couplings make the quantum operations insensitive to cavity-induced loss [28].Thanks to the optimal working points (nk = 0.5), the charge qubits can prolong the coher-ence time by fighting against the 1/f noise. Additionally, the dephasing effects are usuallystronger than the relaxation ones [33], the present encoded scheme is just immune to the σz-type collective dephasing effects, which is helpful to enhance the fidelity. It is found that thedephasing time of the encoded logic qubits may be prolonged to two orders of magnitudelonger using well-designed charge qubits with same device parameters [18, 31].
In conclusion, we have proposed a scheme for performing quantum gates in the chosenDFSs with charge qubits. Qubits are coupled to the one-dimensional TLR which serves asa quantum data bus to virtually exchange photons between qubits. In the dispersive condi-tions, the resonator-assisted interqubit coupling can be accomplished controllably by tuningthe external fluxes only. Based on the switchable coupling, we have implemented the DFS-encoded universal quantum gates. Compared with the non-DFS situation, we have numeri-cally analyzed the robustness of the DFS-encoded gate operations that are insensitive to the
2290 Int J Theor Phys (2012) 51:2282–2290
collective noises. Thus the present scheme provides the potential opportunity of performingthe fault-tolerant quantum computing with Josephson charge qubits.
Acknowledgements We thank Jia-Jun Yang for various helps. This work was supported by the NationalNatural Science Foundation of China under Grant No. 11047006, and No. 11004168.
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