Additivity of decoherence measures for multiqubit quantum systems

Additivity of decoherence measures for multiqubit quantum systems

Physics Letters A 328 (2004) 87–93 www.elsevier.com/locate/pla Additivity of decoherence measures for multiqubit quantum systems Leonid Fedichkin, Ar...

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Physics Letters A 328 (2004) 87–93 www.elsevier.com/locate/pla

Additivity of decoherence measures for multiqubit quantum systems Leonid Fedichkin, Arkady Fedorov, Vladimir Privman ∗ Center for Quantum Device Technology, Department of Physics and Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY 13699, USA Received 6 April 2004; received in revised form 28 May 2004; accepted 2 June 2004 Available online 21 June 2004 Communicated by C.R. Doering

Abstract We introduce new measures of decoherence appropriate for evaluation of quantum computing designs. Environment-induced deviation of a quantum system’s evolution from controlled dynamics is quantified by a single numerical measure. This measure is defined as a maximal norm of the density matrix deviation. We establish the property of additivity: in the regime of the onset of decoherence, the sum of the individual qubit error measures provides an estimate of the error for a several-qubit system. This property is illustrated by exact calculations for a spin-boson model.  2004 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 03.65.Yz Keywords: Decoherence; Fault tolerance; Spin-boson; Qubit; Quantum computer

1. Introduction Dynamics of quantum systems coupled to dissipative environment has long been a subject of study in diverse fields [1–6]. Recent interest in quantum computing has focused attention [7–20] on quantifying environmental effects that cause small deviations from the isolated-system quantum dynamics. During short time intervals of “quantum-gate” functions, environment-induced relaxation/decoherence effects * Corresponding author.

E-mail address: [email protected] (V. Privman). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.06.021

must be kept below a certain threshold in order to allow fault-tolerant quantum error correction [21–28]. The reduced density matrix of the quantum system, with the environment traced out, is usually evaluated within some approximation scheme, e.g., [29–31]. In this Letter, we introduce a new, additive measure of the deviation from the “ideal” density matrix of an isolated system. For a single two-state system (a qubit) this measure is calculated explicitly for the environment modeled as a bath of harmonic modes [32], e.g., phonons. Furthermore, we establish that for a several-qubit system, the introduced measure of decoherence is approximately additive for short times. It can be estimated by summing up the devi-

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ation measures of the constituent qubits, without the need to carry out a many-body calculation, similar to the approximate additivity expected for relaxation rates of exponential approach to equilibrium at large times. In most quantum computing proposals, quantum algorithms are implemented in the following way. Evolution of the qubits is governed by a Hamiltonian consisting of single-qubit operators and of two-qubit interaction terms [33,34]. Some parameters of the Hamiltonian can be controlled externally to implement the desired algorithm. During each cycle of the computation the Hamiltonian remains constant. This ideal model does not include the influence of the environment on the computation, which necessitates quantum error correction. The latter involves non-unitary operations [21–28] and cannot be described as Hamiltoniangoverned dynamics. The accepted approach to evaluate environmentally induced decoherence involves a model in which each qubit is coupled to a bath of environmental modes [32]. The reduced density matrix of the system, with the bath modes traced out, then describes the time-dependence of the system dynamics [7–18, 29–31,35–37]. Because of the interaction with environment, after each cycle the state of the qubits will be slightly different from the ideal. The resulting error accumulates at each cycle, so that large-scale quantum computation is not possible without implementing fault-tolerant error correction schemes [21–28,38–40]. These schemes require the environmentally induced decoherence of the quantum state in one cycle to be below some threshold. Its value, defined for uncorrelated single qubit error rates, was estimated [41,42] to be between 10−6 and 10−4 . To evaluate error rate for a given system, one should first obtain the evolution of its density operator. In realistic cases various approximations are utilized [7–19,29–31,35–37]. The most familiar are the Markovian-type approximations [35–37] used to evaluate approach to the thermal state at large times. It has been pointed out recently that these approximations are not suitable for quantum computing purposes because they are usually not valid [6,19] at low temperatures and for the short cycle times of quantum computation. Several non-Markovian approaches [7–20,29– 31] have been developed to evaluate the system dynamics at short-times.

Additional issues arise when one tries to study decoherence of several-qubit systems. One has to consider the degree to which noisy environments of different qubits are correlated [43]. If all the qubits are effectively immersed in the same bath, then there is a way to reduce decoherence for this group of qubits without error correction procedures, by encoding the state of one logical qubit in a decoherence free subspace of the states of several physical qubits [44–47]. In a large scale quantum computer consisting of thousands of qubits, it is more appropriate to consider qubits immersed in distinct baths, because these errors represent the “worst case scenario” that necessitates error-correction. Once we have the density matrix ρ(t) for a singleor few-qubit system evaluated in some approximation, we have to compare it to the ideal density matrix ρ (i) (t) corresponding to quantum algorithm without environmental influences. We have to define some measure of decoherence, D(t), to compare with the fault-tolerance criteria, which are not specific with regards to the error-rate definition [21–28]. It is convenient to have the measure non-negative, and vanishing if and only if the system evolves in complete isolation. Since explicit calculations beyond one or very few qubits are exceedingly difficult, it would be also useful to find a measure which is additive (or at least sub-additive), i.e., the measure of decoherence of a composite system will be the sum (or not greater than the sum) of the measures of decoherence of its subsystems. In this Letter, we identify two measures of decoherence which have these desirable properties, and one of which is also relatively easy to calculate analytically and is not sensitive to the frequencies of the system’s internal dynamics. We prove the asymptotic additivity and derive explicit results for decoherence in a short-time approximation. For entangled qubits, such properties have not been established for the traditional measure of degree of deviation from a pure state by quantum von Neumann entropy [48], nor for the fidelity measure described, e.g., in [49–53]. In Section 2, we introduce our preferred operator-norm measure of decoherence. The additivity of both decoherence measures within for a spinboson type model is explored in Sections 3 and 4. An extended version of our results has been presented in [54].

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2. Operator norm as a measure of decoherence Typically, the Hamiltonian of a quantum system interacting with environment has the form H = HS + HB + HI ,

(1)

where HS is the internal Hamiltonian, HB is the Hamiltonian of the environment (bath), HI is the system-bath interaction term. Over gate-function cycles, the overall density matrix R(t) of the system and bath evolves according to R(t) = e−iH t /h¯ R(0)eiH t /h¯ .

(2)

Usually the initial density matrix R(0) = ρ(0) ⊗ Θ

(3)

is assumed [35–37] to be a direct product of the initial density matrix of the system, ρ(0), and the thermalized density matrix of the bath, Θ. The reduced density matrix of the system is obtained by tracing out the bath modes, ρ(t) = TrB R(t).

(4)

There have been several approaches considered in the literature to quantify the deviation of the density matrix of the system from the ideal (without environment) dynamics [49–53,55]. In order to introduce our operator-norm measure, we start with the deviation, σ (t) = ρ(t) − ρ (i) (t),

(5)

of the reduced density matrix from the ideal ρ (i) (t) ≡ e−iHS t /h¯ ρ(0)eiHS t /h¯ .

(6)

As a numerical measure of the deviation we will use the operator norm σ λ . It is defined [56] as the maximal absolute value of the eigenvalues of the matrix σ . For a 2 × 2 deviation matrix, 1/2  . σ λ = |σ11 |2 + |σ12 |2 (7) Note that σ λ is not only a function of time, but it also depends on the initial density operator ρ(0). Due to decoherence, it will increase from zero as time goes by. However, its time-dependence will also contain oscillations at the frequencies of the internal system dynamics, as illustrated in Fig. 1. The effect of the bath can be better quantified by the maximal deviation

Fig. 1. Schematic plot of the deviation norm for a two-level system interacting with the Ohmic bath [1–5] of bosonic modes in the short-time approximation [19]. The oscillatory curves illustrate the σ λ norms for three different initial states ρ(0). The monotonic curve shows the D norm, obtained as σ λ maximized (8) over ρ(0). Note that in this case D(∞) = 1/2.

norm,     D(t) = sup σ t, ρ(0) λ .

(8)

ρ(0)

This measure of decoherence, which is always in [0, 1], will typically increase monotonically from zero at t = 0, saturating at large times at a value D(∞)  1. The definition of the maximal decoherence measure D(t) looks rather complicated for a general multiqubit system. However, in what follows we show that it can be evaluated in closed form for short times, appropriate for quantum computing, for a single-qubit (two-state) system. We then establish an approximate additivity that allows us to estimate D(t) for severalqubit systems as well. To be specific, let us outline the calculation of this measure of decoherence of a two-level system (spin, qubit) interacting with a bosonic bath of modes. Thus, we take h¯ Ω σz , HS = − 2  † hω HB = ¯ k ak ak , k

HI = σx

  gk ak† + gk∗ ak . k

(9) (10) (11)

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Here ωk are the bath mode frequencies, ak , ak† are the bosonic annihilation and creation operators, gk are the coupling constants, σx and σz are the Pauli matrices, and hΩ ¯ > 0 is the energy gap between the ground (up, |1) and excited (down, |2) states of the qubit. The dynamics of the system can be obtained in closed form in the short-time approximation [19]. The general expression [19] for the density operator of a spin-1/2 system is  ρpq (0)m|µµ|pq|νν|n ρmn (t) = p,q=1,2 µ,ν=±1

× ei(Eq +En −Ep −Em )t /(2h¯ )−B

2 (t )(η

µ −ην )

2 /4

. (12) Here the Roman-labeled states, |j , with j = m, n, p, q = 1, 2, are the eigenstates of HS corresponding to the eigenvalues Ej /h¯ = (−1)j Ω/2. The Greeklabeled states, |ζ , are the eigenstates of σx , with the eigenvalues ηζ = ζ , where ζ = µ, ν = ±1. The spectral function B 2 (t) is defined according to  B 2 (t) = 8 |gk |2 (h¯ ωk )−2 k

× sin (ωk t/2) coth(βωk /2). 2

(13)

Here β = h/k ¯ BT ,

(14)

kB is the Boltzmann constant, and T is the initial bath temperature. This function has been studied in [6,44,57]. With ρ12 (0) = |ρ12 (0)|eiφ ,

(15)

we get σ (t)λ =

2  1 2 1 − e−B (t ) ρ11 (0) − ρ22 (0) 2  2   1/2 + 4ρ12 (0) sin2 (Ω/2)t + φ .

(16)

In Fig. 1, we show schematically the behavior of σ (t)λ for three representative choices of the initial density matrix ρ(0). Generally, σ (t)λ increases with time, reflecting the decoherence of the system. However, oscillations at the system’s internal frequency Ω are superimposed, as seen explicitly in (16). The effect of the bath is better quantified by the maximal operator norm D(t). In order to maximize

over ρ(0), we can parameterize   ρ(0) = U p|11| + (1 − p)|22| U † ,

(17)

where 0  p  1, and U is an arbitrary 2 × 2 unitary matrix,

i(α+γ ) e cos θ ei(α−γ ) sin θ U= (18) . −ei(γ −α) sin θ e−i(α+γ ) cos θ By using the rigorous definition [56] of the norm, ψ|A† A|ψ , ψ|ψ ψ=0

A2λ = sup

(19)

one can show that in this case it suffices to put p = 1 and search for the maximum over the remaining three real parameters α, γ and θ . The result, sketched in Fig. 1, is D(t) =

 1 2 1 − e−B (t ) . 2

(20)

3. Subadditivity properties of the decoherence measures In quantum computing, the error rates can be significantly reduced by using several physical qubits to encode each logical qubit [45–47]. Therefore, even before active quantum error correction is incorporated [21–28], evaluation of decoherence of several qubits is an important, but formidable task. Consider a system consisting of two initially unentangled subsystems S1 and S2 , with decoherence norms DS1 and DS2 , respectively. We denote the density matrix of the full system as ρS1 S2 and its deviation as σS1 S2 , and use a similar notation with subscripts S1 and S2 for the two subsystems. If the evolution of the system were governed by the “non-interacting” Hamiltonian of the form H S1 S2 = H S1 + H S2 ,

(21)

where the terms HS1 , HS2 act only on variables of the system S1 , S2 , respectively, then the overall norm DS1 S2 could be bounded by the sum of the norms DS1 and DS2 ,  (i)  DS1 S2 = sup σS1 S2 λ = supρS1 S2 − ρS1 S2 λ ρ(0)

ρ(0)

 (i) (i)  = supρS1 ⊗ ρS2 − ρS1 ⊗ ρS2 λ ρ(0)

L. Fedichkin et al. / Physics Letters A 328 (2004) 87–93

  = supσS1 ⊗ ρS2 + ρS(i) ⊗ σS2 λ 1

Consider again the composite system consisting of the two subsystems S1 , S2 , with the “non-interacting” Hamiltonian (21). The evolution superoperator of the system will be

ρ(0)

    sup σS1 ⊗ ρS2 λ + supρS(i) ⊗ σ S 2 λ 1 ρ(0)

ρ(0)

 sup σS1 λ + sup σS2 λ = DS1 + DS2 . ρS1 (0)

ρS2 (0)

(22) In general, even for interacting subsystems, initially unentangled qubits will remain nearly unentangled for short times, because they didn’t have enough time to interact. Therefore, for nearly unentangled qubits, the inequality DS1 S2  DS1 + DS2

(23)

is expected to provide a good approximate estimate for DS1 S2 , i.e., the measures for individual qubits can be considered approximately additive. Establishing a similar approximate bound for entangled qubits is considerably more complicated, and will be accomplished in Section 4. For large times, the separate measures become of order 1, so this bound is not useful. Instead, the rates of approach of various quantities to their asymptotic values are approximately additive in some cases [43,55]. In the rest of this Letter, we therefore focus on the short-time regime and establish a much stronger property: as mentioned, we prove the approximate additivity for initially entangled qubits whose dynamics is governed by (9)–(11) with coupling to independent baths. For intermediate calculations, we use the “diamond” norm [38–40], which we denote by K,   K(t) = T − T (i)     = sup T − T (i) ⊗ IG  . (24) 

Tr

Here, the trace norm [56] is given by ATr = Tr A† A.

(25)

The two superoperators are defined as generating the actual and “ideal” dynamical evolutions, T (t) : ρ(0) → ρ(t),

91

T (i) (t) : ρ(0) → ρ (i) (t),

(26) IG is the identity superoperator on a Hilbert space G whose dimension is the same as that of the corresponding space of the superoperators T and T (i) , and  is a density operator in this space of twice the number of qubits.

TS1 S2 = TS1 ⊗ TS2 ,

(27)

and the ideal one will be TS(i) = TS(i) ⊗ TS(i) . 1 S2 1 2

(28)

The diamond measure for the whole system can be expressed [38–40] as  (i)  KS1 S2 = TS1 S2 − TS1 S2       =  TS1 − TS(i) ⊗ TS2 − TS(i) ⊗ TS2 + TS(i) 1 1 2   (i)     TS1 − TS1 ⊗ TS2  (i)  (i)  + TS1 ⊗ TS2 − TS2       TS  = TS1 − TS(i) 2 1  (i)   (i)     + TS1 TS2 − TS2    (i)  (i)  = TS − T  + TS − T  1

S1

= KS1 + KS2 .



2

S2



(29)

The step that follows the inequality, in (29), proved in [38–40] as the “stability” property, was established specifically with the definition (24). The approximate inequality KS1 S2  KS1 + KS2

(30)

for the diamond norm K has thus the same form as for the norm D. Let us emphasize, however, that there is an important difference in that the relation for D further requires the subsystems to be initially unentangled. This restriction does not apply to the relation for K. This property is particularly useful for quantum computing, which is based on qubit entanglement. However, even in the simplest case of the diamond norm of one qubit, the calculations are extremely cumbersome. Therefore, the measure D is favorable for actual calculations.

4. Derivation of the additivity property and summary To recapitulate, the measure K explored in the preceding section allows derivation of stronger results.

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However, the measure of decoherence D is significantly easier to calculate. Therefore, we aim at establishing stronger results for D for the spin-boson model in the short-time approximation by relating the two measures. While in general one can prove that 2D(t)  K(t)  2,

(31)

for a single qubit, explicit calculation within the shorttime approximation actually gives

It follows that

 

 1 2 2 Bj + o Bj = Dj + o Dj , D 2 j j j j (39) where we used (20) for short-times. By combining the upper and lower bounds, we get the final result for short-times,

  Dj (t) + o Dj (t) . D(t) = (40) j

K(t) = 2D(t)

(single qubit).

(32)

Since the multiqubit norm D is generally bounded by K/2, it follows that for the specific model considered, with a bosonic heat bath acting independently on each of the qubits, the multiqubit D is approximately bounded by the sum of the single-qubit norms even for the initially entangled qubits,  K  Kj  1 − e−Bj  = = Dj , D 2 2 2 2

j

j

(33)

This completes the proof of additivity of the measure D(t) at short times, for spin-qubits interacting with bosonic environmental modes. This result should apply as a good approximation up to intermediate, inverse-system-energy-gap times, e.g., [19], and it is consistent with the recent finding [58] that at short times decoherence of a trapped-ion quantum computer scales approximately linearly with the number of qubits.

j

where j = 1, . . . , N labels the qubits. For short times, one can also establish a lower bound on D(t). Consider a specific initial state with all the qubits excited, ρ(0) = (|22|)1 ⊗ · · · ⊗ (|22|)N .

(34)

Then according to (12), ρ(t) = ρ1 (t) ⊗ · · · ⊗ ρN (t),

Acknowledgements This research was supported by the National Security Agency and Advanced Research and Development Activity under Army Research Office contract DAAD19-02-1-0035, and by the National Science Foundation, grant DMR-0121146.

(35) References

where ρj (t) =

 2 1  1 − e−Bj (t ) |11| 2  −B 2 (t )  + 1 + e j |22| .

(36)

The right-bottom matrix element of the (diagonal) deviation operator,

  2 σ2N ,2N (t) = −1 + 2−N (37) 1 + e−Bj (t ) ,

[1] [2] [3] [4] [5] [6] [7]

j

can be expanded as

 1 2 Bj (t) + o Bj2 (t) , 2 j

j

[8] [9] [10]

(38)

j

because B(t) is linear in t for small times. The largest eigenvalue of σ (t) cannot be smaller than σ2N ,2N (t).

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