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Journal of the Korean Physical Society, Vol. 37, No. 5, November 2000, pp. 496∼502

Dynamical Suppression of the Decoherence in Quantum Registers Koungtae Kim∗ and Hyuk-jae Lee

†

Institute of Quantum Information Processing and Systems, University of Seoul, Seoul 130-743

Doyeol Ahn‡ Department of Electrical Engineering, University of Seoul, Seoul 130-743

Sung Woo Hwang Permanent address: Department of Electronics Engineering, Korea University, Seoul 136-701 (Received 16 June 2000) We have studied the collective decoherence dynamics of a quantum register coupled with a thermal environment. It is shown that by applying a spin-flipping radiofrequency pulse sequence, the Lamb phase shift, as well as the phase damping, of the quantum register become controllable and the coherence of the quantum register is recovered.

I. INTRODUCTION

the evolution of a quantum open system can be modified by applying an externally controllable interaction. That work was based on eliminating the unwanted effects of the environmental interaction by applying a proper sequence of pulses. Although the phenomenon is mathematically similar to the quantum Zeno effect, the essential idea behind the physics is completely different. Carr and Purcell proposed a clever scheme for eliminating classically the effect of the magnetic diffusion term in spin echoes [6], and the authors in Ref. [4] addressed whether similar techniques would work in the presence of a quantum-mechanical environment. The latter authors started with a simple model with a qubit coupled to a thermal bath of harmonic oscillators and showed that decoherence was dynamically suppressed by repeating effective time-reversal operations on the total system. They assumed that the typical decoherence time of the system and the duration of the pulses defined two different time scales. Here, we consider a more general model with an `-qubit interacting with the common environment collectively [7–9] and address the question of whether a similar decoherence correction mechanism will work in an `-qubit model under the same condition. For our purpose, in the following section, we investigate the decoherence properties for an `-qubit open quantum system. Section III describes the method for applying the controllable pulsed perturbation and the scheme for reducing the unwanted decoherence effects as a limiting situation. Finally, we mention some concluding remarks and suggestions.

Quantum computers that are designed to be perfectly isolated from their environments are in principle, able to solve hard, computationally demanding problems more powerfully than conventional classical computers [1] can. For a real quantum system, however, one can not completely neglect the interaction with its environment. This interaction causes the problem known as decoherence, which corrupts the information stored in the quantum system and produces wrong outputs. To perform a quantum computation we must prepare qubits in a proper initial state and keep them free from interactions that lead to noise and decoherence. Thus, we have to overcome such difficulties to realize practical quantum computers. There have been proposals for more realistic quantum computers. One is to use error-correcting codes encoded in a linear subspace of the total Hilbert space in such a way that errors induced by the interaction with environment can be detected and corrected [2]. Another is to use the error-avoiding codes that are barely corrupted rather than states that can be easily corrected. These error-avoiding codes can be constructed by building a nontrivial subspace of the total Hilbert space that allows a weaker correlation with the environment than the remaining part of the Hilbert space does [3]. Alternatively, recent works [4,5] have demonstrated that ∗

E-mail: [email protected] E-mail: [email protected] ‡ E-mail: [email protected] †

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Dynamical Suppression of the Decoherence in Quantum Registers – Koungtae Kim et al.

II. COLLECTIVE DECOHERENCE The quantum decoherence of an open system is a purely quantum-mechanical process which entangles the quantum state with the environmental degrees of freedom [10]. It is the fact that we do not have any knowledge about the environment that destroys the coherence of the quantum state. Physically, decoherence is an unwanted noise due to scattering, diffusion, and localization in the Hilbert space of the quantum system. Since the development of quantum information processing crucially depends upon maintaining the coherence of the quantum system, we will investigate how the decoherence properties of an open quantum system may be modified through the application of an external controllable interaction. To start, we introduce a physical system which is ` identical qubit systems (`-qubit register) jointly coupled with a single environment modeled by a bath of non-interacting oscillators. The `-qubit can be described by ` spin-1/2 systems. The coupling between the system and the bath spreads out the quantum information encoded over the system Hilbert space into the combined system plus bath Hilbert space and results in decoherence of the qubits. We assume that the dynamics of the total system are governed by H = HS + HB + HI , where HS =

` X 1 i=1

2

0 σiz , HB =

HI =

X

i=1

(gk a†k + gk∗ ak )σiz .

k

P`

1 z i=1 2 σi

z

S = being global spin operators which provide a representation of the Lie algebra su(2); [S α , S β ] = iαβγ S γ .

(4)

The time evolution from t0 to t of the total system is determined by the time-ordered unitary operator Z t ˆ (t, t0 ) = T exp − i ˆ ) . U dτ H(τ (5) t0

This evolution operator can be evaluated exactly as follows. Since the Hamiltonians at different times do not ˆ (t, t0 ) as commute with each other, we cannot have U Rt ˆ )]. a direct exponentiation of the form exp[−i t0 dτ H(τ Using the fact that the qubit operators are constants of the motion and exploiting Wick’s theorem, which gives a relation between time-ordering and normal-ordering of ˆ (t, t0 ) as operators [11], we can decompose U (6)

a†k (ak )

(1)

k

Here, the σiz ’s are the usual Pauli operators (i is the qubit index), and a†k and ak are bosonic creation and annihilation operators for the field mode with wave number k, respectively. HS (HB ) is the Hamiltonian of the register (bath), and HI is the register-bath interaction. Here, we assume that the coupling constants, gk , to the environment are the same for all qubits and that the mutual interactions between the qubits are sufficiently small so that we can safely neglect them. Throughout the paper, we use units in which ¯h = 1. The state of the total system S + B is described in full generality in terms of a density operator ρT (t). We then consider the reduced density operator of qubits: ρS (t) = TrB [ρT (t)],

dition, since the model in Eq. (1) is exactly soluble, it allows a clear picture of the decoherence properties. It is convenient to move into the interaction picture associated with the free dynamics HS + HB , where the effective Hamiltonian is given as X ˆ ˆ I = 2S z H(t) =H (gk a†k eiwk t + gk∗ ak e−iwk t ), (3)

ˆ (t, t0 ) = Vˆ (t, t0 )W ˆ (t, t0 ), U

wk a†k ak ,

k

` X X

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(2)

where TrB denotes a partial trace over the degrees of freedom of the environment. In the Hamiltonians in Eq. (1), σiz is a conservative operator, so the coupling with the environment in this model gives a pure decoherence process [7]. For full generality, we have to consider the quantum dissipation which arises from energy exchanges between the register and the bath. However, we can neglect the quantum dissipation because the energyexchange processes typically involve time scales much longer than those of decoherence mechanisms. In ad-

ˆ ) only contains where Vˆ (W operators of the bath. This is possible due to the fact that the qubit-part operators commute at all times. The Vˆ (t, t0 ) operator is defined by i

X dVˆ = 2S z gk a†k Vˆ , dt

(7)

i,k

ˆ (t, t0 ) will be determined from the evolution of and W ˆ (t, t0 ), which is derived from the the equation for U Schr¨ odinger equation ˆ dU ˆ I (t)U ˆ. =H dt We get the solution of Eq. (7) as ( ) X † z iw t Vˆ (t, t0 ) = exp S a e k 0 ξk (t − t0 ) , i

k

(8)

(9)

k

where 2gk (1 − eiwk t ). (10) wk With these and the operator relation exp(−ξk a†k )ak exp(ξk a†k ) = ak + ξk , we have the equation ˆ: for W X ˆ dW ˆ i = 2S z ak gk∗ e−iwk t W dt k X ˆ, + 2(S z )2 gk∗ ξk (t − t0 )e−iwk (t−t0 ) W (11) ξk (t) =

k

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Journal of the Korean Physical Society, Vol. 37, No. 5, November 2000

where each |αi i(i= 1...`) is an eigenstate of σiz with eigenvalue αi = ±1 so that |αi is an eigenstate of P` P` S z = i=1 12 σiz with eigenvalue Sαz = i=1 12 αi . Then, the initial density operator of the system, ρS (t0 ), can be expanded in terms of the computational basis as

and the solution is ( ) X ˆ (t, t0 )=exp −S z W ak e−iwk t0 ξk∗ (t−t0 ) k

(

z 2

×exp −(S )

X |ξk (t−t0 )|2 k

2

z 2

)

+i(S ) φ(t−t0 ) ,

(12) ρS (t0 ) =

with φ(t) =

X

4|gk |2

k

wk t − sin(wk t) . wk2

(16)

(13)

k

We suppose that the system and the environment are initially uncorrelated with each other so that the density operator of the total system is ρT (t0 ) = ρS (t0 ) ⊗ ρB (t0 ). Furthermore, the environment is assumed to be in thermal equilibrium at a temperature T so that

ρB (t0 ) = (14)

The factor exp[i|S z |2 φ(t)] is the Lamb phase shift, which plays an important role in the collective decoherence, unlike in the single-qubit system where the Lamb shift only gives an overall phase factor [8,12]. The time evolution of the system (register) is comˆ (t, t0 ). To show pletely determined by the operator U this explicitly, let us introduce a computational basis of qubits: |αi = |α1 i ⊗ |α2 i ⊗ · · · ⊗ |α` i,

ρ(t0 )αβ |αihβ|.

α,β

Collecting all these results gives the following evolution operators: ˆ (t, t0 ) U ( X † =exp S z ak eiwk t0 ξk (t−t0 )−ak e−iwk t0 ξk∗ (t−t0 ) o 2 +i S z φ(t − t0 ) .

X

(15)

Y

ρB,k (T ) =

k

Y e−HB,k /kB T k

ZB,k

,

(17)

where HB,k (ZB,k ) is the Hamiltonian (partition function) of the k-th oscillator in the thermal bath and kB is the Boltzmann constant. We have used the fact that at thermal equilibrium the density operator of the bath is given as a direct product of the density operator of each of its modes. Now the reduced density operator of qubits can be exactly evaluated by tracing out the bath degrees, and we get

h i ˆ (t, t0 )ˆ ˆ † (t, t0 ) ρˆS (t) = TrB U ρS (t0 ) ⊗ ρˆB (t0 )U X Y = ρˆ(t0 )αβ |αihβ| Trk ρB,k D eiwk t0 ξ(t − t0 )(Sαz − Sβz ) α,β

k

× exp i (Sαz )2 − (Sβz )2 φ(t − t0 ) X = ρˆ(t0 )αβ |αihβ| exp −(Sαz − Sβz )2 Γ(t − t0 ) exp i (Sαz )2 − (Sβz )2 φ(t − t0 ) ,

(18)

α,β

where D(ξk ) = exp(ξk a†k − ξk∗ ak ) is a displacement operator of the coherent state, whose thermal average value for each mode is given by [13] Trk [ρB,k D(ξk )] = exp 12 |ξk |2 coth(wk /2kB T ) (19) and the phase damping factor Γ(t) =

X |ξk (t)|2 k

2

coth(wk /2kB T ).

(20)

The meanings of the phase damping factor and the Lamb phase shift term can be clearly seen from the reduced density operator, Eq. (18), of the system. There are no

changes in the populations of the states of the system whereas there is a decay of the coefficients of the offdiagonal elements of the density operator and the phase shift. This loss of coherence arises because of the ignorance of the environment after tracing over the bath modes. By calculating the fidelity defined by F(t) = Tr[ˆ ρS (t)ˆ ρS (t0 )], we can measure the degree of decoherence in the qubits [14]. For an initially prepared pure state |ΨS i = |ΨS (t0 )i, the fidelity is reduced to F(t) = hΨS (t0 )|ˆ ρS (t)|ΨS (t0 )i. In our case, it reads

(21)


F(t) =

X α,β

|hα|ΨS ihΨS |βi|2 exp −(Sαz − Sβz )2 Γ(t − t0 ) cos (Sαz )2 − (Sβz )2 φ(t − t0 ) .

This clearly shows that the phase damping term causes the decay of the fidelity and that the Lamb phase shift contributes to the oscillatory behaviors. Another measure of the coherence of a quantum state is the linear entropy, which is defined by L(t) = Tr[ˆ ρ(t) − ρˆ(t)2 ] [15] and which measures the purity of a time-evolved quantum state that was initially prepared as a pure state. After some calculations we get X L(t)=1− |hα|ΨS ihΨS |βi|2 exp −2(Sαz−Sβz )2 Γ(t−t0 ) . α,β

(23) The purity does not involve the Lamb phase shift.

III. PULSED EVOLUTION OF QUANTUM COHERENCE Next we will consider modifications of the decoherence properties of the quantum register. Those modifications are obtained by applying the controllable pulsed perturbation Hrf (t). The procedure we will use is explained in Ref. [4]; application of a sequence of π pulses allows the unwanted effects of the coupling, HI , between the system and the environment to be averaged out. That work considered only a single qubit coupled to the environment, so the Lamb phase shift term only gave an overall phase factor. However, as in our case, the Lamb phase shift term, in addition to the phase damping, is an important source of decoherence in a multi-qubit system. We assume that Hrf (t) is zero except for times which are very short compared to the typical decoherence time. During these very short times, Hrf (t) is so large that HI can be neglected in comparison. This idealization enables us to say that Hrf produces spin rotations at the time the pulse is applied, but that, between pulses, the total system develops under the action of H [16]. Specifically, the pulse Hamiltonian is assumed to be a train of identical N π-pulses applied on resonance at instants t = τn with pulse separation ∆t: Hrf =

N X

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2Vn (t)[cos(0 t)S x + sin(0 t)S y ],

(24)

n=1

where τn = t0 + n∆t, n = 1, 2, ..N , and V τ n ≤ t ≤ τ n + tw , Vn (t) = 0 otherwise,

In order to depict the evolution associated with a given pulse sequence, we find it convenient to think of the latter as being formed by successive elementary cycles of spinflips, a complete cycle being able to return the spin back to the starting configuration. We begin by calculating the time-evolution operator during the first spin cycle consisting of two π-pulses. In the interaction picture, the Hamiltonian with the pulse sequences (H(t) + Hrf (t)) becomes ˆ ˆ I (t) + H ˆ rf (t), H(t) =H

(27)

ˆ rf (t) is evaluated by using Eq. (24) and the propwhere H erties of Pauli matrices. We get ˆ rf (t) = H

N X

2Vn (t)S x .

(28)

n=1

In the interaction picture, the time dependence due to the rotating field in Hrf is completely removed within each pulse. The effect of a pulse of amplitude V and duration tw at time τn is to produce a transformation by the unitary operator Pî : x Pî = e−2iV tw S ,

(29)

where V tw is chosen as π/2 in the limits of Eq. (26) so that Pî = −i

` Y

σix .

(30)

i=1

Let us calculate the evolution operator corresponding to a two-pulse cycle consisting of the following sequence: ˆ I (t) during t0 ≤ t ≤ τ1 ; pulse Pˆ1 at evolution under H ˆ I (t) during τ1 ≤ t ≤ τ2 ; and time τ1 ; evolution under H pulse Pˆ2 at time τ2 . After a total time t1 = τ2 = t0 + 2∆t, one cycle is complete, and the evolution operators corresponding to one cycle can be written as ˆp (t1 , t0 ) = Pˆ2 U ˆ (τ2 , τ1 )Pˆ1 U ˆ (τ1 , t0 ) U i h ih ˆ (τ2 , τ1 )Pˆ1 U ˆ (τ1 , t0 ) . (31) = Pˆ2 Pˆ1 Pˆ1−1 U The unitary evolution operator between the pulse sequences is obtained from Eq. (14) as ˆ (τ2 , τ1 ) U = exp{S z

(25)

X

a†k eiwk τ1 ξk (∆t)−ak e−iwk τ1 ξk∗ (∆t)

k

z 2

with tw being the width of the pulse and V its height. In the following we work under the ideal condition tw → 0, V → ∞, V tw → finite,

(22)

(26)

+ i|S | φ(∆t)}, the product of the two pulse-evolutions is Pˆ2 Pˆ1 = −1,

(32)

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and ˆ (τ2 , τ1 )Pˆ1 Pˆ1−1 U X =exp{−S z a† eiwk (t0+∆t) ξk (∆t)−ak e−iwk (t0+∆t) ξk∗ (∆t) k

z 2

+i|S | φ(∆t)}.

The next step is to generalize the description to an arbitrary number N of elementary spin-flip cycles in which the nth cycle ends at time tn = t0 + 2n∆t,

n = 1, . . . , N ,

(36)

(33)

The final expression for the evolution operator for the first cycle (t0 → t1 = t0 + 2∆t) now can be written as ˆp (t1 , t0 ) U ( X † =exp S z ak eiwk t0 ηk (∆t)−ak e−iwk t0 ηk∗ (∆t) k

+i|S z |2 φ(2∆t) ,

(34)

where

2 2gk 1 − eiwk t . (35) ηk (t) = wk Now, we can clearly see the effects of the pulses by comˆ (t1 , t0 ) and U ˆp (t1 , t0 ) paring the evolution operators U in Eqs. (14) and (34), respectively. There is no change in the Lamb phase shift term during one spin-flip cycle. However, the coefficient of the bosonic operators changes to ηk (∆t) ∝ (ξk (∆t))2 in place of ξk (2∆t). Below we will notice that this change modifies the decoherence properties of the quantum register and induces a modification of the Lamb phase shift.

the number of π-pulses involved in the full sequence being 2N . It is straightforward to write down the evolution operator for the nth cycle from knowledge of the firstcycle evolution operator given in Eq. (34): ˆp (tn , tn−1 ) U X † =exp{S z ak eiwk tn−1 ηk (∆t)−ak e−iwk tn−1 ηk∗ (∆t) k

+|S z |2 φ(2∆t)}.

(37)

The total evolution operator corresponding to N cycles is then governed by the time-ordered finite product of the evolution operator for each cycle: ˆp(N ) (tN , . . . , t0 ) = U

N Y

ˆP (tn , tn−1 ). U

(38)

n=1

By exploiting the recursion relations, we can find a closed ˆp(N ) : form of U

Xh i † iwk t0 (N ) z −iwk t0 ∗ z 2 ˆ Up = exp S ak e ηk (N, ∆t) − ak e ηk (N, ∆t) + i|S | φp (N, 2∆t) ,

(39)

k

where 1 − e2iwk N ∆t ηk (N, ∆t) = ηk (∆t) , 1 − e2iwk ∆t 1 X N sin(2wk ∆t) − sin(2N wk ∆t) φp (N, 2∆t) = |ηk (∆t)|2 + N φ(2∆t). 2 1 − cos(2wk ∆t)

(40) (41)

k

Comparing Eq. (39) with Eq. (14) enables us to see the effects of the pulse sequence explicitly. The final density operator of the register corresponding to the pulsed evolution is obtained by tracing out the bath degrees of freedom through the steps outlined for the unperturbed case. The expression for the density operator at time tN = t0 + 2N ∆t is X ρˆS (tN ) = ρˆS (t0 )αβ |αihβ| exp −(Sαz−Sβz )2 Γp (N, ∆t) α,β

where

× exp i (Sαz )2 − (Sβz )2 φp (N, 2∆t) ,

Γp (N, ∆t) =

X |ηk (N, ∆t)|2 k

2

coth(wk /2kB T ).

(42)

Notice that ρˆαα (tN ) = ρˆαα (t0 ). Even with the spin-flip pulse sequences the spin populations are unchanged in the final density operator. A comparison with Eq. (20), shows that the modification of the decoherence properties in the presence of pulses is quiet straightforward: just use ηk (N, ∆t) instead of ξk (tN − t0 ) for each mode of the bath and φp (N, 2∆t) instead of φ(tN − t0 ) for the Lamb phase shift. From the expressions for Γp (N, ∆), and φ(N, 2∆t) it is not easy to figure out the contribution of the pulse sequence to the suppression of decoherence. Let us consider an idealized situation represented by the following limits which describe continuous spin-flip pulse sequences

(43) ∆t → 0, N → ∞, 2N ∆t = tN − t0 .

(44)


In this limit, 2

lim |ηk (N, ∆t)|

∆t→0

=

4|gk |2 |1−eiwk (tN −t0 ) |2 |1 − eiwk ∆t |4 lim =0. (45) 2 ∆t→0 |1 − e2iwk ∆t |2 wk

If this result holds for an arbitrary field mode, then the implications for the decoherence properties are transparent: lim Γp (N, ∆t) = 0.

∆t→0

(46)

Furthermore, for the Lamb phase shift, we have lim φp (N, 2∆t) = 0.

∆t→0

(47)

Equations (46) and (47) clearly show that in the limit of continuous flipping, the decoherence and the Lamb shift are completely eliminated, regardless of the temperature and the spectral density function of environment. In the laboratory, such a mathematical continuous limit can not be implemented. However, notice that for each mode k, |ηk (N, ∆t)|2 wk ∆t 2 = tan , (48) |ξk (tN − t0 )|2 2 so if we apply pulse sequences with a separation ∆t, which is sufficiently shorter than the time scale determined from the cut-off frequency wc of the spectral density of the bath, the pulsed evolution of qubits suffer less phase damping. Moreover, in the limit wc ∆t 1, the Lamb phase shift in the presence of pulses is also sufficiently reduced: |φp (N, 2∆t)| 2 X 2 wk (tN−t0 )−sin(wk (tN−t0 )) wk ∆t ≈ 4|gk | wk2 2 k 2 wc ∆t ≤ |φ(tN − t0 )| . (49) 2 These are our main results: the decoherence in the quantum register can be reduced by applying a pulse sequence with a sufficiently narrow pulse width.

IV. CONCLUSIONS We have studied the decoherence in a quantum register which consists of ` noninteracting identical qubits but which is interacting with a thermal bath. The quantum state of the register is subject to phase damping, in which the off-diagonal elements of the density operator decay, and to Lamb phase shifting of the quantum state, which is an important source of decoherence in a multiqubit system like a quantum register. In order to realize a quantum computer, we have to eliminate this decoherence. In a previous work [4], the authors addressed how

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the phase damping in a single qubit can be protected by introducing a controllable external interaction to the quantum register but we showed how to generalize their model to an `-qubit system. In this work we showed that introduction of a controllable external interaction to the quantum register can control the phase damping and the Lamb phase shift and that in the limit of continuous applications, all these sources of decoherence can be eliminated completely so that an exact unitary evolution of the quantum system can be recovered. Although our results depend upon the integrability of the model explicitly, we expect that there may be more general procedures for preserving the quantum states and the quantum gates from decoherence, which are important ingredients of a quantum computer [17]. We, therefore, believe firmly that our system is suitable for practical quantum computing.

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