OSA's Digital Library

Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 2, Iss. 9 — Apr. 27, 1998
  • pp: 372–377
« Show journal navigation

Characterization of decoherence processes in quantum computation

J. F. Poyatos, J. I. Cirac, and P. Zoller  »View Author Affiliations


Optics Express, Vol. 2, Issue 9, pp. 372-377 (1998)
http://dx.doi.org/10.1364/OE.2.000372


View Full Text Article

Acrobat PDF (491 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We show how the dynamics of open quantum systems can be fully characterized by using quantum tomography methods. We apply these methods to the case of an ion trap quantum computer, which does not operate under ideal conditions due to coupling to several environments. We study the performance of a fundamental two–bit quantum gate as a function of various parameters related to the interaction of the ions with external laser fields.

© Optical Society of America

1. Introduction

2. Characterizing open quantum systems

Here, we briefly discuss the general formalism to characterize the dynamics of an open quantum system. For a more detailed discussion we refer the reader to Ref. 3

3. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html [CrossRef]

. Suppose we are trying to describe a given physical process in a particular system. Because of the interaction with the environment this process is generally described by

ρ̂inερ̂out=ε[ρ̂in],
(1)

where ε is the linear superoperator that must conserve trace, positivity, and self-adjointness. The goal is thus to characterize ε. If the system is initially prepared in a pure state |Ψin〉 = Σi=0N ci |i〉 where |i〉 are the basis vectors spanning the (N + 1)-dimensional input space, after the evolution the state of the system is given by

ρ̂out=i,i'=0Nci[ci']*R̂i'i,
(2)

In order to obtain the operators i′i , one must perform the following series of experiments. First, one must prepare the system in a given set of initial states, and, second, one must perform state tomography in the final states. It is now interesting to know how close the physical implementation of our ideal process is to the real one. In order to study this, we introduce several parameters. First, we could be interested in knowing how close the final state after the process is to the ideal one. For that purpose, we introduce the Fidelity F

F=ΨinÛρ̂outÛΨin¯,

Where the overline indicates the average over all possible input states |Ψin〉, and Û is the unitary operator corresponding to the ideal process. One could also study the effect of the decoherence. To this aim, we introduce the Purity 𝛲

𝑝=Tr{(ρ̂out)2}¯,
(3)

which is averaged as before. Both parameters can be calculated once the tomography of the process has been implemented.

We will apply this formalism to the particular example of a two–bit gate. We will then consider only nonentangled states as input states and local single measurements in the output states; otherwise we would need to apply a two–bit quantum gate in the preparation and measurement, which would mask the whole procedure.

3. The trapped ion quantum gate

We now apply the above ideas in the context of the dissipative ion trap quantum computer model. First, we explain in more detail some of the requirements previously mentioned, namely the set of initial states needed and the measurements associated with the state tomography. As was stated before, we will consider only nonentangled states as input states. As an example, the initial states needed in this case can be given by the 16 product states |ψa1|ψb2 (a, b = 1,…, 4), where

ψ1=0,ψ3=12(0+1),
ψ2=1,ψ4=12(0+i1).
(4)

The quantum tomography of the output states can be carried out following the lines proposed by Wooters [6

6. William K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1-21, (1987) [CrossRef]

]. Writing the density operator as

ρ̂out=q=015λqÂq,
(5)

where Âq = σ^ q1 1σ^ q2 2 (q = 4q 1 + q 2), with σ^ q1 a = {1̂a, σ^ x a, σ^ y a, σ^ z a}, and a = 1, 2 refers to the first and second qubits, respectively. By measuring the observables Âq , one can determine the coefficients λq , given that λq = Tr[ρ^ out Âq ]/4. It is important in this context that neither for the preparation of the initial states nor for the tomographic measurement are two–bit quantum gates necessary. Since it is precisely the two–bit quantum gate that we want to characterize, it is pointless to use as tools for this characterization, other two–bit quantum gates. On the other hand, single qubit measurements and operations are considered to be error free.

We have considered two ions in a linear ion trap interacting with two lasers. Let us denote by |gn = |0〉n and |en = |1〉n two internal states of the nth ion, and by |e′)n an auxiliary internal state. As we have shown in Ref. 3

3. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html [CrossRef]

, the universal two–qubit gate defined by

1122(1)121122,(1,2=0,1),
(6)

can be implemented in three steps: (i) Apply a π laser pulse to the lower motional sideband corresponding to the transition |g1 → |e1 of the first ion; (ii) apply a 2π laser pulse to the lower motional sideband of the transition |g2 → |e′2 of the second ion; (iii) apply a π laser pulse, as in (i). By lower motional sideband we mean that the laser frequency must be equal to the corresponding internal transition frequency minus the trap frequency in order to excite a center of mass phonon only. An alternative way of performing conditional gates with trapped ions beyond the requirement of cooling to zero temperature has been recently developed [7

7. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Computations with Trapped Ions at non-zero Temperature,” (unpublished).

]. In our description, we will also consider the presence of dissipation in the phonon modes, the most important source of dissipation in realistic experiments. The interaction of the two ions and the laser is given by the following master equation:

ρ˙=i[H,ρ]+ρ
(7)

where

H=Δ1e11eΔ2e'22e'+νacmacm+3νarar
+Ω1(t)2[e11geiηcm(acm+acm)eiηr(ar+ar)+H.c.]
+Ω2(t)2[e'22geiηcm(acm+acm)eiηr(ar+ar)+H.c.],
ρ=κcm(2acmρacmacmacmρρacmacm)
+κr(2arρarararρρarar).
(8)

Here, ∆1,2 and Ω1,2 are the laser detunings and Rabi frequencies, respectively, of the laser acting on each ion. The operators a and a are annihilation and creation operators of the center of mass (cm) and relative (r) motion mode, η is the corresponding Lamb-Dicke parameter, κ is the phonon dissipation rate, and υ is the trap frequency. Dissipation has been described by means of the standard quantum optics formalism of master equations, based on the Born–Markov and rotating wave approximations, and, by considering a linear coupling between the phonon modes and the reservoir of external modes at zero temperature.

With a numerical calculation we have simulated the measurement of the operators i′i . The idea is that by working only in the system space we are able to study the realistic gate that we are performing (see Fig. 1).

Fig. 1. System environment scheme in a realistic process.

Apart from the previously introduced Fidelity and Purity, we introduce also the so-called “Quantum Degree of the Gate” 𝑄, defined as the maximum value of the overlap between all possible output states that are obtained starting from an unentangled state and all the maximally entangled states, i.e.,

𝑄=maxρ˜out,|ΨmeΨmeρ˜outΨme,

where ρ˜ out denote the output states corresponding to unentangled input states |Ψin〉 = |ψa1|ψb2, and |Ψme〉 is a maximum entangled state. As has been shown, when the overlap between a density operator and a maximally entangled state is larger than (2 + 3√2)/8 ≃ 0.78, Clauser-Horne-Shimony-Holt inequalities are violated [8

8. C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996) [CrossRef] [PubMed]

]. Finally, another useful parameter is the “Entanglement Capability” 𝐶 [9

9. A. Peres, “Separability Criterion for Density Matrices,” Phys. Rev. Lett. 77, 1413 (1996). http://xxx.lanl.gov/abs/quant-ph/9604005 [CrossRef] [PubMed]

,10

10. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

], given as the smallest eigenvalue of the partial transposed density matrix ρ^ out, for unentangled inputs states. As recently shown [6

6. William K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1-21, (1987) [CrossRef]

], the negativity of this quantity is a necessary and sufficient condition for nonseparability of density operators of two spin 1/2 systems. These quantities can be calculated numerically starting from the gate operators i′i with a maximization/minimization procedure. Both parameters are important to quantify the ability to create entanglement between the two ions. In Fig. 2 we have plotted the different gate parameters as functions of the dissipation rate. As expected, the gate becomes unreliable whenever κtg ≃ 1, where tg is the time required for the gate that is of the order of 4π /(Ωη). We note also that the purity of the gate decreases faster compared to that of the fidelity, which in turn decays in the same way as the degree of entanglement.

Fig. 2. Fidelity, Purity Quantum degree of a Gate and Entanglement Capability as functions of the dissipation rate. Here κ cm/υ=κ r/υ=κ/υ, η=l and Ω/υ=.l and ∆ = -υ.

In Fig. 3 we have plotted the fidelity of the gate as a function of the laser detuning. Under ideal circumstances, the gate should work for a detuning ∆ = -υ. As we see, when the effective Rabi frequency (proportional to Ωη/υ) increases, the Fidelity decreases. The maximum Fidelity occurs for a different value of ∆. This is due to the AC-Stark shift, which is a consequence of the laser radiation.

Fig. 3. Fidelity as a function of the detuning. Here we have chosen Ω/υ=.l, κ/υ=0.

4. Conclusions

J. F. P. acknowledges University of Innsbruck for its hospitality. This research was supported in part by the European TMR network ERB-FMRX-CT96-0087.

Footnotes

also with Departamento de Física Aplicada, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

References and links

1.

A. Ekert, “From quantum-codemakingto quantum code-breaking,” Report No. quant-ph/9703035, (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997), and references therein. http://xxx.lanl.gov/abs/quant-ph/9703035

2.

D. P. Divincenzo, “Topics in Quantum Computers,” Report No. cond-mat/9612126 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1996), and references therein. http://xxx.lanl.gov/abs/cond-mat/9612126

3.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html [CrossRef]

4.

J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091, (1995). [CrossRef] [PubMed]

5.

I. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” Report No. quant-ph/9610001 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997).http://xxx.lanl.gov/abs/quant-ph/9610001

6.

William K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1-21, (1987) [CrossRef]

7.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Computations with Trapped Ions at non-zero Temperature,” (unpublished).

8.

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996) [CrossRef] [PubMed]

9.

A. Peres, “Separability Criterion for Density Matrices,” Phys. Rev. Lett. 77, 1413 (1996). http://xxx.lanl.gov/abs/quant-ph/9604005 [CrossRef] [PubMed]

10.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

OCIS Codes
(000.6800) General : Theoretical physics
(270.0270) Quantum optics : Quantum optics

ToC Category:
Focus Issue: Control of loss and decoherence in quantum systems

History
Original Manuscript: November 14, 1997
Published: April 27, 1998

Citation
Juan Poyatos, Ignacio Cirac, and Peter Zoller, "Characterization of decoherence processes in quantum computation," Opt. Express 2, 372-377 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-9-372


Sort:  Journal  |  Reset  

References

  1. A. Ekert, "From quantum-codemaking to quantum code-breaking," Report No. quant-ph/9703035, (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997), and references therein. http://xxx.lanl.gov/abs/quant-ph/9703035
  2. D. P. Divincenzo, "Topics in Quantum Computers," Report No. cond-mat/9612126 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1996), and references therein. http://xxx.lanl.gov/abs/cond-mat/9612126
  3. J.F. Poyatos, J.I. Cirac and P. Zoller, "Complete Characterization of a Quantum Process: The two-bit Quantum Gate," Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html
    [CrossRef]
  4. J. I. Cirac and P. Zoller, "Quantum Computations with Cold Trapped Ions," Phys. Rev. Lett. 74, 4091, (1995).
    [CrossRef] [PubMed]
  5. I. Chuang and M. A. Nielsen, "Prescription for experimental determination of the dynamics of a quantum black box," Report No. quant-ph/9610001 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997). http://xxx.lanl.gov/abs/quant-ph/9610001
  6. William K. Wootters, "A Wigner-function formulation of finite-state quantummechanics," Ann. Phys. 176, 1-21, (1987)
    [CrossRef]
  7. J. F. Poyatos, J. I. Cirac and P. Zoller, "Quantum Computations with Trapped Ions at non-zero Temperature," (unpublished).
  8. C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. Wooters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Phys. Rev. Lett. 76, 722, (1996)
    [CrossRef] [PubMed]
  9. A. Peres, "Separability Criterion for Density Matrices," Phys. Rev. Lett. 77, 1413 (1996). http://xxx.lanl.gov/abs/quant-ph/9604005
    [CrossRef] [PubMed]
  10. M. Horodecki, P. Horodecki, and R. Horodecki, "Separability of Mixed States: Necessary and Sufficient Conditions," Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited