Evaluating holonomic quantum computation beyond adiabatic limitation

a r X i v :q u a n t -p h /0208120v 1 18 A u g 2002

Evaluating holonomic quantum computation:beyond adiabatic limitation

LiXiang Cen 1,2,XinQi Li 1,2,YiJing Yan 2,HouZhi Zheng 1,and ShunJin Wang 3

1

NLSM,Institute of Semiconductors,The Chinese Academy of Sciences,Beijing 100083,People’s Republic of China

2

Department of Chemistry,Hong Kong University of Science and Technology,Kowloon,Hong Kong

3

Department of Physics,Sichuan University,Chengdu 610064,People’s Republic of China

The proposal of the optical scheme for holonomic quantum computation is evaluated based on dynamical resolution to the system beyond adiabatic limitation.The time-dependent Schr¨o dinger equation is exactly solved by virtue of the crank-ing representation and gauge transformation approach.Be-sides providing rigorous con?rmation to holonomies of the geometrical prediction that holds for the ideally adiabatic situation,the dynamical resolution enables one to evaluate elaborately the amplitude of the nonadiabatic deviation,so that the errors induced to the quantum computation can be explicitly estimated.

The recently proposed holonomic approach to quan-tum computation [1]-[4]surely predicts a striking contri-bution to the application of quantum physics.Transcend-ing the traditional dynamical means for quantum compu-tation,the holonomic approach realizes quantum infor-mation processing by endowing the quantum code with a non-trivial global topology (a gauge ?eld potential)and the associated holonomies then allow for the universal quantum computing.Speci?cally,in the scheme of holo-nomic quantum computation,information is encoded in a degenerate eigenspace of the governing Hamiltonian and the holonomies (abelian as well as non-abelian)[5]-[7]are acquired by driving the system to undergo appropri-ate loops in the parameter space adiabatically.Besides suggesting an intriguing connection between the gauge ?elds and the information processing,such a geometri-cal means for quantum manipulation is believed to have built-in fault-tolerant features [8]-[11]due to its inher-ent stability against local perturbations.Considerable attention has been addressed to this topic recently and the all-geometrical implementation for universal quan-tum gates has been proposed by optical schemes,based on laser manipulation of ions con?ned in a Paul trap [12]or neutral atoms in an optical resonator [13].

The existent exploration for holonomic quantum com-putation is based on the analysis by pure geometrical fashion.It is true that in the adiabatic limit the holon-omy associated with the evolving loop is determined by the path traced by the time-evolution ray and the cur-vature of the ray space.This involves the abelian holon-omy (the Berry phase)and the non-abelian one merely known as adiabatic connection.Nevertheless,as a whole physical problem,as the dynamics of a system generates a time-dependent physical state,a speci?ed geometrical object (the ray)is generated as well.In such a sense,

dynamics determines the holonomy through determining the ray itself and its path.Moreover,in view that the re-alistic evolution of a physical system could not be ideally adiabatic and the nonadiabaticity shall alter the time-evolution of the ray and thus inevitably induce deviation from the adiabatic consequences.The evaluation of such deviation and the resulting errors in quantum computa-tion is de?nitely a dynamical problem that goes beyond the geometrical exploration.

In this paper we employ a tractable model of the op-tical scheme to exploit this subject.For the appropri-ately chosen loops of the Hamiltonian in the parameter space,the time-dependent Schr¨o dinger equation is ex-actly solved by virtue of the cranking representation and gauge transformation approach.The derived dynami-cal evolution of the system recovers the holonomic trans-formation provided by geometrical consequences,includ-ing the simple abelian phase factor and the general non-abelian operation.Thus our results provide further con-?rmation of the geometrical prediction,and besides,the errors caused by nonadiabatic e?ects for the holonomic quantum gate operation can be estimated explicitly.For the proposed optical scheme of holonomy quantum computation [12,13],the basic idea relies on the adiabatic passage via the dark states since the dynamical evolution restricted to such a space is completely trivial.The sys-tem encoding the qubit is realized by a four-level Λ-type trapped ion (or a similar cavity atom).The three ground levels |g i (i =1,2,3)are highly degenerate and each cou-ples to the excited state |e in a tunable way.The states |g 1 and |g 2 stand for the computational bases |0 and |1 ,respectively,and |g 3 is an ancillary level required for implementation of gate operations.Such a system admits two dark states that have no contribution from the excited state.Through changing the Rabi frequen-cies and driving the dark states to undergo appropriate cyclic evolutions in an adiabatic fashion,the universal single-bit gate operations e iφ|1 1|and e iφσy can be gen-erated due to the global geometry of the bundle of the eigenspace of the dark states.

To evaluate the gate operation e iφ|1 1|from a dynami-cal viewpoint,let us explore the state evolution generated by the periodic Hamiltonian [12,13]

H (t )=?sin θ(σ2e +σe 2)+?cos θ(σ3e e i?+σe 3e ?i?),

(1)

where θis a ?xed parameter and ?is assumed to rotate at a constant frequency γfor convenience.The equation of motion for the system is

1

i

?

dt

=

?I (t )

?)

=?x i (γ

?

)=| ψ0(0)|D (0) |2=| ψ0(T )|D (T ) |2.

(8)

Besides the leakage,the nonadiabatic evolution shall result in deviation to the desired phase factor.It fol-lows,instead of the net Berry phase,the cyclic evolution here induces a total phase (the so-called Lewis-Riesenfeld phase)

Φ= T

0 ψ0(t )|i ?

γ.(9)

The detailed depiction of the deviation for the phase fac-tor is shown in Fig.

1(b).Noting that in the adiabatic

limit,the total phase [15]

Φ=lim γ/?→0

2π

x 0(γ

γ/?=4πsin 2θ,(10)

the geometrical consequence is thus recovered.

The validity of the above evaluation is based on a presumption that the initial state |D (0) =cos θ|g 2 ?sin θ|g 3 can be generated from the computational ba-sis |g 2 and so the inverse process.Explicitly,such pro-cesses can be accomplished by the driven Hamiltonian (1)through changing the parameter θadiabatically.Con-ventionally,the nonadiabatic e?ect here shall lead to an additional error for the quantum computation.However,such an error can be in principle avoided through ap-pending a matching interaction to compensate the gauge potential term induced to the system.Speci?cally,one can use the following Hamiltonian (setting ?=0)H tot (t )=H (t )+H ad (t ),H ad (t )=i ˙θ

(t )(σ23?σ32).(11)

It follows that the dynamical invariant of the system

H tot (t )now has a form I (t )=H (t ),thus the above state transformation can be processed exactly.Physically,the interaction H ad (t )can be realized by a microwave cou-pling to the two degenerate levels |g 2 and |g 3 ,with its intensity accurately controlled through a derivative feed-back process.

Now we investigate the gate operation e iφσy achieved by the holonomic means.The corresponding evolution is generated by the Hamiltonian

H (t )=?sin θcos ?(σ1e +σe 1)

+?sin θsin ?(σ2e +σe 2)+?cos θ(σ3e +σe 3),(12)where the parameter ?=γt .As is known,the adi-abatic cyclic evolution of the Hamiltonian generates a non-abelian holonomy due to its degeneracy structure of the dark states.It can be easily worked out,from the formula of Ref.[7],that the holonomic transformation

u C =e i 2πcos θD y ,

(13)

where D y =i (|D 2 D 1|?|D 1 D 2|),and the two dark states,|D 1 =cos θ|g 1 ?sin θ|g 3 and |D 2 =|g 2 ,span the degenerate space of the starting (ending)Hamilto-nian.Note that the Hamiltonian (12)possesses an su(4)Lie algebraic structure and dynamical resolution to the system is usually very complicated.Surprisingly,as we shall show in the following,this system can be exactly solved by the gauge transformation approach [16,17],and its dynamical evolution analytically manifested thus leads to a complete understanding of the adiabatic and nonadiabatic properties for the time-dependent Hamilto-nian system.

Similar to the cranking method used above,we intro-duce the unitary gauge transformation

U g (t )=e ?γt (σ12?σ21)

(14)

2

to the equation of motion for the system,from which a covariant Schr¨o dinger equation is stemmed

|Ψg(t) =U?1g(t)|Ψ(t) ,

i

?

?t

=?sinθ(σ1e+σe1)

+?cosθ(σ3e+σe3)+iγ(σ12?σ21).(16) In view that the above Hamiltonian is time independent, the basic solutions|Ψn g(t) to the covariant equation(15) can be easily obtained and the corresponding eigenvalues are as follows

E1,2=±√

2

ˉ? 1? ?)2cos2ˉθ 1/2,

E3,4=±√

2

ˉ? 1+ ?)2cos2ˉθ 1/2,(17)

where

ˉ?=? 1+(γ/?)2.(18)

The dynamical basis of the system(12)can be directly obtained as|Ψn(t) =U g|Ψn g(t) ,from which one can see that E n has the natural implication related to the total phase.Now the time evolution operator generated by the Hamiltonian(12)can be given

U C(T)=

4

n=1|Ψn(T) Ψn(0)|

=

4

n=1e?iE n2π

γ

=

±cosθ,and the phase-equipped dynamical bases|Ψ1(t) and|Ψ2(t) have the form

|Ψ1(t) =√

2

e?iγt cosθ[(cosθcosγt+i sinγt)|g1 +(cosθsinγt?i cosγt)|g2 ?sinθ|g3 ],

|Ψ2(t) =√

2

e iγt cosθ[(cosθcosγt?i sinγt)|g1

+(cosθsinγt?i cosγt)|g2 ?sinθ|g3 ].(20)

One can verify that they are the instantaneous eigen-states of the Hamiltonian(12)with a two-degeneracy eigenvalue0,and the equipped phases are just the Berry phases accordingly.Thus the cyclic evolution restricted to the space spanned by these two states is purely geo-metrical and can be denoted as

u(T)=e?i2πcosθ|Ψ1(0) Ψ1(0)|+e i2πcosθ|Ψ2(0) Ψ2(0)|

(21) with|Ψ1(0) =(|D1 ?i|D2 )/√

2.It can be easily recognized that the opera-tor(21)is just the non-abelian holonomy(13),thus the geometrical nature is veri?ed again.

The above dynamical resolution to the system is im-portant.Besides o?ering a vivid veri?cation to the re-markable formula of non-abelian holonomy[7],which holds for the ideally adiabatic situation,it enables one to evaluate elaborately the amplitude of the nonadiabaticity deviation and the resulting errors to the holonomic gate operation e iφσy.In detail,the population transfer from the initial state|Ψ(0) =|g2 is pictured in Fig.2.The leakage out of the computational space can be estimated by the projection(see also Fig.2)

η(θ,

γ

?1??2are tunable.One can see that the bases|g1g1 , |g1g2 ,and|g2g1 are decoupled from the evolution,and

the component|g2g2 serving as the code|11 evolved in an enclosed space spanned by{|g2g2 ,|g3g3 ,|ee }.In-troducing the su(3)generators explicitly

A e2=e i2?1|ee g2g2|,A e3=e i2?1|ee g3g3|,

A23=|g2g2 g

3

g3|,A?μν=Aνμ,(25)

the Hamiltonian(24)can be rewritten as

H eff=g sinθ(A2e+A e2)+g cosθ(A3e e i?+A e3e?i?).

(26) Obviously this Hamiltonian possesses an su(3)algebraic structure isomorphic to that of system(1),thus all the

discussions therein also hold for the present system.

It should be noted that,the e?ective Hamiltonian(24),

respecting a second-order process of the interaction,is quite a rough description of the model.Speci?cally,it ignores the same second-order process induced by virtual

photons excitation in the self-transitions of the states: |g2(3) →|g2(3) and|e →|e .It can be anticipated that such self transitions shall dress the energy levels of the ions and lift the degeneracy of the ground states,

which in turn a?ects the desired gate operation.Detailed exploration of this point shall be presented in a future report.

This work was supported in part by the Postdoctoral

Science Foundation,the special funds for Major State Basic Research Project No.G001CB3095,the National Natural Science Foundation No.10175029of China,and the Research Grants Council of the Hong Kong Govern-ment.

?

) determined by Eq.(7).

[16]S.J.Wang,F.L.Li and A.Weiguny,Phys.Lett.A180,

189(1993).

[17]S.Wang,B.Hu,Q.Jie,and B.Li,Algebraic dynam-

ics,gauge transformation,and geometric phase(private

communication).

[18]K,Molmer and A.Sorensen,Phy.Rev.Lett.82,1835

(1999).

Caption of Fig.1:

Deviation induced by nonadiabaticity for abelian holonomy.(a)The overlapηfor the parametersθ∈(0,π/2)andγ/?∈[0,1];(b)The total phaseΦfor θ∈[0,π]andγ/?∈[0,1].

Caption of Fig.2:

Deviation induced by nonadiabaticity for non-abelian holonomy.The initial state is prepared in|D2 .The two solid curves show the results for the population of the target state on|D1 and|D2 ,as a function of1?cosθ,respectively.The dashed curve depicts the total populationηon the computational space.Figures(a),

(b),(c)and(d)correspond toγ/?=0.01,0.2,0.5and

0.8,respectively.

4

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