# The Rabi Oscillation in Subdynamic System for Quantum Computing

Authors: Bi Qiao; Gu Jiayin
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

A quantum computation for the Rabi oscillation based on quantum dots in the subdynamic system is presented. The working states of the original Rabi oscillation are transformed to the eigenvectors of subdynamic system. Then the dissipation and decoherence of the system are only shown in the change of the eigenvalues as phase errors since the eigenvectors are fixed. This allows both dissipation and decoherence controlling to be easier by only correcting relevant phase errors. This method can be extended to general quantum computation systems.


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## Body

## 1. Introduction

The Rabi oscillation based on the quantum dot system is an efficient way to realize quantum logic operation [1]. The projected formalism of control rotation gate (CROT) for two qubits is expressed as negative CNOT operation [2]. The physical observation of the Rabi oscillation with 10π period has been reported, and the detail analysis of quantum computation based on the Rabi oscillation and relevant mechanism of decoherence and dissipation have been discussed by several works with various methods [3–12]. However, the controlling dissipation and decoherence in the Rabi oscillation system is still an important problem to challenge application of the Rabi oscillation in quantum computation.On the other hand, subdynamics theory rooted from the old Brussels-Austin school [13–15] and followed by some up-to-date works [16, 17] has been found that there is a potential advantage to cancel both decoherence and dissipation in quantum information systems. This is so because the relevant subdynamic kinetic equation (SKE) intertwines with the original Schrödinger equation or Liouville equation by a similarity transformation, which allows to construct the relative ideal subdynamic system in which the eigenvectors are fixed when decoherence or dissipation happens. Therefore the errors will only exist in the phase change of the evolution of projected states which makes easier to cancel the decoherence and dissipation.Considering this background, in this work, we present a subdynamic proposal [13–20] to realize quantum computing based on the Rabi oscillation. This proposal can provide a relative ideal Rabi oscillation used in quantum computing by controlling both dissipation and decoherence. Below the formalism is firstly introduced.


## 2. Subdynamic Formalism

Let us consider a two-level system described by a Hamiltonian:(1)H^t=ω0σ00+ω1σ11+12gσ10e-iωLt+σ01e+iωLt, where ω0 and ω1 is an eigen-frequency of states 0 and 1, respectively; σ00=00, σ11=11, σ01=01, and σ10=10; ωL is a frequency of exciting optical field; g=-μE0 is a Rabi frequency between states 0 and 1 jumping, μ is a dipole moment between 0 and 1 jumping, and Et is an envelope of exciting optical field, Et=E0cos⁡ωLt=1/2E0e-iωLt+e+iωLt. Notice the so-called rotation wave approximation has been used to produce above Hamiltonian. Then, for simplicity, here the explicitly time-dependent Hamiltonian can be transformed to the time-independent one by a time-dependent unitary transformation U00t=exp-iωLtσ00, where σ00=00 is a projector [21]. So we obtain a time-independent Hamiltonian as(2)H=U00-1tH^tU00t=ω0σ00+ω1σ11+12gσ01+σ10=H0+H1, where defining(3)H0=ω0σ00+ω1σ11,H1=12gσ01+σ10. When the exciting optical field takes off, U00t=1, we have H^t=H. So, the corresponding Schrödinger equation for the Hamiltonian in (2) is(4)i∂ψ∂t=Hψ,which can reflect the same dynamics described by the Hamiltonian in (1).If considering the influence of environment, the total Hamiltonian can be supposed asH=H0+H1+ΔH, where ΔH comes from the interaction between the system and environment, which may introduce dissipation and decoherence in the system. Then a main frame to construct the subdynamic systems is, firstly to establish a subdynamic equation [13–19], as(5)i∂∂tψ~=Θψ~,where ψ~ is a projected state, H+ΔH=ΩΘΩ-1, and Ω is a similarity operator. These parameters and relations can be constructed by the following several rules.(1) Propose projectors as(6)Pnk=n⊗φkn⊗φk,forn=0,1,k=0,1,2,…,and Qnk=1-Pnk, where n corresponds to the index of eigenvectors for Hamiltonian in system, k denotes the index to the Hamiltonian in environment, and φk is the kth eigenvector of diagonal part of the interaction Hamiltonian between the system and environment.(2) Introduce the creation (destruction) operator expressed as(7)Cnk=PnkHPnk+PnkΔHQnk1Enk-QnkH+ΔHQnkQnkΔHPnk,Cnk†=Dnk.(3) Introduce the projected state in the subdynamic system as(8)ψ~=Ω-1ψ, where ψ is a solution of the original Schrödinger equation and the similarity operator Ω is defined by(9)Ωnk=Cnk+Pnk,Ω=∑n=01∑kΩnk,Ω-1=∑n=01∑kΩnk-1=∑n=01∑kPnk+DnkCnk-1Pnk+Dnk.(4) The ruling equation is subdynamic Liouville equation:(10)i∂∂tρ~=Θ,ρ~, where Θ operator is introduced by(11)Θ=∑n=01∑kΘnk=∑n=01∑kPnkHPnk+PnkΔHQnkCnkPnk, and the relationship of the above Θ operator with the original H+ΔH can be proven by a kind of similarity transformation:(12)Θ=Ω-1H+ΔHΩ. So it is obvious to use (12) and consider (8); one can obtain SKE (5) from the Schrödinger equation or the Liouville equation.


## 3. Rabi Oscillation in Subdynamic Space

Using the Rabi oscillation between1 (or 11) and 0 (or 00) as the working states of quantum computation is a reasonable approach of quantum computing based on quantum dot system [11, 22]. In the ideal situation, ΔH=0, the Liouville equation for this two-level system with the Rabi oscillation is expressed by(13)iU11-1t∂ρ∂tU11t=U11-1tH,ρU11t=U11-1tHρU11t-U11-1tρHU11t=U11-1tU11tH^tU11-1tρU11t-U11-1tρU11tH^tU11-1tU11t=H^tU11-1tρU11t-U11-1tρU11tH^t=H^t,ρt=i∂U11-1tρU11t∂t=i∂ρt∂t, where one notices that using (13) one has(14)∂U11-1t∂tρU11t+U11-1tρ∂U11t∂t=0,ρ=U11tρtU11-1t=ρ1111+ρ0000+ρ1010+ρ0101.Then, from (13) one gets the time-independent equations of nρm=ρnm as(15)iddtρ11=12gρ01-ρ10,iddtρ00=-iddtρ11=-12gρ01-ρ10,iddtρ10=12gρ00-ρ11. Then by introducing U, V, W as a Bloch vector, that is,(16)U→=Ue^1+Ve^2+We^3 with(17)U=ρ10+ρ01,V=iρ10-ρ01,W=ρ11-ρ00, where e^1, e^2, e^3 is an unit vector, respectively, and e^i·e^j=δij, thus (15) can be changed as(18)ddtUVW=00000g0-g0UVW.Then using the subdynamic similarity operator Ω above, one gets(19)
Ω
-
1
d
U
d
t
d
V
d
t
d
W
d
t
=
Ω
-
1
0
0
0
0
-
0
g
0
-
g
0
Ω
Ω
-
1
U
V
W
,
(20)
d
d
t
U
~
V
~
W
~
=
Θ
1
0
0
0
Θ
2
0
0
0
Θ
3
U
~
V
~
W
~
.The solution of (20) is given by(21)ΩU~tV~tW~t=Ω-ig000ig0000Ω-1ΩU~0V~0W~0,where(22)UVW=ΩU~V~W~=001i-i0110U~V~W~,so (20) can also give a solution based on the Bloch ball,(23)UtVtWt=1000cos⁡θ-sin⁡θ0sin⁡θcos⁡θU0V0W0, where θ is defined as a radial rotation angle. In the initial condition (by considering (17) which corresponds to an initial pure state of the Rabi oscillation ρ00=1 and else three states, ρ11, ρ01, and ρ10 as zero), one gets(24)U0V0W0=00-1.This allows one, from (23), to get(25)UtVtWt=0-sin⁡θt-cos⁡θt,which reflects the Rabi oscillation in the ideal situation without the interaction from environment.


## 4. Controlling Dissipation and Decoherence

Now let us consider there is an interaction between the system and environment. This can introduce some dissipation and decoherence. For example, an interaction of Liouvillian for the Rabi oscillation system can be specified by(26)ΔLρ=12γ102σ01ρσ10-σ11ρ-ρσ11,where σ10 or σ01 expressed the same meaning as in (1) and γ10 is a coupling number reflecting dissipation velocity. So the corresponding master equation for the pseudovector ρ=ρ00,ρ11,ρ01,ρ10 is expressed as(27)ddtρ11=-i12gρ01-ρ10-γ10ρ11,ddtρ00=-ddtρ11=i12gρ01-ρ10+γ10ρ11,ddtρ10=-iδρ10-i12gρ00-ρ11-12γ10ρ10; then through the subdynamic similarity operator Ω one obtains(28)iddtΩ-1UVW=Ω-1-1T2-δ0δ-1T2g0-g-1T1ΩΩ-1UVW+Ω-100-1T1,where we notice that δ=ω1-ω0-ωL; T2=1/2T1+γph, T1=1/γ10 expresses the life time of state 1, and γph is a coupling number reflecting pure phase decoherence. Then considering (18) one can define(29)L+ΔLUVW=-1T2-δ0δ-1T2g0-g-1T1UVW+00-1T1. This allows us to introduce the subdynamics for U,V,W→ρ=ρ00,ρ11,ρ22,ρ01,ρ10 or ρ~=ρ~00,ρ~11,ρ~22,ρ~01,ρ~10 by the above formalism:(30)ddtΩ-1ρt=Ω-1L+ΔLΩΩ-1ρt=Θρ~t=ddtρ~t with(31)Θ=Ω-1L+ΔLΩ=PL+ΔLP+PL+ΔLQCP,ρt=Ωρ~t=P+Cρ~t=∑nmPnm+Cnmρ~t, by defining the projector(32)P=∑nmPnm=∑nmρ~nmρ~nm,fornm=00,11,01,10, with(33)Q=1-P,so that ρ=ρ00,ρ11,ρ01,ρ10 is expressed by the subdynamic formalism:(34)ρnm=Pnm+Cnmρ~nm.This ρnm can be measured in the practical system with energy term lnm, and then using ρnm, lnm we can construct the projected states ρ~nm in the subdynamic space by the subdynamic formalism:(35)ρnm=ρ~nm+Qnm1Mnn-QnmL+ΔLQnmQnnL+ΔLρ~nn,where it is marvelous that whatever the system has L or L+ΔL, the above formalism gives the same ρ~=ρ~00,ρ~11,ρ~22,ρ~01,ρ~10→U~,V~,W~ in the subdynamic space; namely, if L of the computing system changes to L+ΔL then the similarity operator Ω in (22) changes to(36)
Ω
=
-
1
δ
g
δ
g
δ
g
0
1
g
-
δ
2
-
g
2
-
1
g
-
δ
2
-
g
2
1
1
1
,which, from the changed U,V,W, still gives the same U~,V~,W~ as(37)U~V~W~=-δgδ2+g20δ2δ2+g2δg2δ2+2g212g-δ2-g2g22δ2+2g212δgδ2+g2-12g-δ2-g212g2δ2+g2UVW.Hence this equation group allows one, from the measurement of states ρ=ρ00,ρ11,ρ01,ρ10 and energy lnm, for nm=00,11,01,10, to determine the projected states ρ~=ρ~00,ρ~11,ρ~01,ρ~10. These projected states are invariant in the subdynamic space in the sense:(38)Θρ~nm=ρ~nmHρ~nm+ρ~nmHQnmCnmρ~nm=θnmρ~nm,Θρ~nm=ρ~nmH+ΔHρ~nm+ρ~nmH+ΔHQnmCnmρ~nm=θnm′ρ~nm,in which both cases have the same eigenvectors ρ~nm→U~,V~,W~, although the eigenvalues are changed from θnm to θnm′. So from (21) one can obtain(39)-ig000ig0000⟶-1T2--δ2-g2000-1T2+-δ2-g2000-12g2T2g2+δ2e-it/T2-1, where one notices that the evolution W~ is given by(40)iddtW~=-1T2W~-12g2T2g2+δ2,W~t=-e-∫1/T2dt∫12g2T2g2+δ2e∫1/T2dt+C=-e-it/T2∫0ti2g2T2g2+δ2ei1/T2tdt=12g2T2g2+δ2e-it/T2-1. Hence based on the above subdynamic formalism the physical realizing states in the subdynamic space are related to U~,V~,W~ which can be determined by the states U,V,W.The remaining change of the eigenvalue fromθnm to θnm′, which includes two parts of influence of the dissipation and decoherence, may introduce the relative phase error. However since the eigenvector in the subdynamic space is fixed, the correction procedure only needs to consider phase error; this makes the procedure easier; for instance, suppose the working states in quantum computing process are designed as ρ~00,ρ~11,ρ~01,ρ~10; then the phase error correction in the evolution of U~t,V~t,W~t is only necessarily considered. This can be given by using (39): (41)U~tV~tW~t=ei1/T2+-δ2-g2000ei1/T2--δ2-g2000e-i1/2g2/T2g2+δ2e-it/T2-1U~V~W~,so the correction is to replace -1/T2∓-δ2-g2 by ∓g2 and replace 1/2g2/T2g2+δ2e-it/T2-1 by 0, which enables (41) to return (21). For example, using the quantum three-editing-code method [23] if there are phase errors in W~W~W~ for a quantum computing tolerance time Δt then one can correct the errors by using(42)xyzx⊕zy⊕zerror-qubitW~W~W~W~W~0W~′W~W~?W~1W~W~′W~W~?2W~W~W~′??3,where ? means W~′⊕W~≠W~ because W~′ has a phase error. Thus by means of ? one can conclude which qubit has error. This process is described by an error point formula: xyzW~W~→xyz,x⊕z,y⊕z. Then the correction is to replace W~′ by W~ through replacing the relevant eigenvalues after considering change of the eigenvalues. This process can keep the working states W~to be corrected by canceling the time evolution phase error of the states during every error tolerance time interval Δt in the subdynamic spaces.Although the above Rabi oscillation seems to deal with one qubit system, it is not problem to generalize to multiqubits systems. For example, it can be applied into a two-qubit system based on a quantum dot with two excitons in which there are four degenerate energy levels, where00 is vacuum state of excitons and 10 and 01 are x or y polarization of exciton states, respectively, while 11 is states of two excitons. Thus using x and y polarization light, the CROT for two qubits can be realized as(43)Ucrot00,01,10,11=00,01,-11,10 with(44)Ucrot=10000100000-10010, which gives an operation on the states based on (17) as(45)UcrotUUVVWW=UUVVWW′, so that WW changes to WW′. Therefore in the subdynamic space one can perform the CROT operation for two-qubit system ideally; that is,(46)
U
crot
U
~
U
~
V
~
V
~
W
~
W
~
=
U
crot
0
-
1
2
i
1
2
0
1
2
i
1
2
1
0
0
U
V
W
0
-
1
2
i
1
2
0
1
2
i
1
2
1
0
0
U
V
W
′
=
U
~
U
~
V
~
V
~
W
~
W
~
′
.The relevant errors can be corrected using the same approach as the previous subdynamic treatment, as in (41) and Table (42).


## 5. Conclusions and Remarks

The subdynamic system for the quantum computation based on the Rabi oscillation is introduced. The working states of subdynamic system are the eigenvectors ofΘ=ΩHΩ-1 which is similarity operator of the original Hamiltonian or Liouvillian. These eigenvectors constructed are invariant in the subdynamic space, so they are independent on the interaction part of Hamiltonian or Liouvillian between system and environment. While the dissipation and decoherence induced by the interaction part of Hamiltonian or Liouvillian exist in the change of the eigenvalues, there are only phase errors in the subdynamic space. This makes easier for one to find a procedure to cancel the phase error since the eigenvectors are fixed, such as the quantum three-editing-code method listed in this work. Moreover, these corrections of phase errors contain not only an error induced by decoherence but also an error induced by dissipation; therefore this method has an advantage compared with other approaches which only deal with the decoherence. Furthermore, this method principally can be extended to general quantum computing system without restriction. For example, if the system uses laser pulses instead of a continuous laser, it still can find the subsystem dynamics where the eigenstates are constant. This is generally true for subdynamics.



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*Source: 151690-2015-11-04.xml*