Advances in Mathematical Physics
Volume 2015 (2015), Article ID 151690, 7 pages
http://dx.doi.org/10.1155/2015/151690

Research Article

The Rabi Oscillation in Subdynamic System for Quantum Computing

Bi Qiao and Gu Jiayin

Department of Physics, Science School, Wuhan University of Technology, Wuhan 430070, China

Received 16 August 2015; Revised 3 October 2015; Accepted 5 October 2015

Academic Editor: Pavel Kurasov

Copyright © 2015 Bi Qiao and Gu Jiayin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 10 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:

where and is an eigen-frequency of states and , respectively; , , , and ; is a frequency of exciting optical field; is a Rabi frequency between states and jumping, is a dipole moment between and jumping, and is an envelope of exciting optical field, . 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 , where is a projector [21]. So we obtain a time-independent Hamiltonian as

where defining

When the exciting optical field takes off, , we have . So, the corresponding Schrödinger equation for the Hamiltonian in (2) is

which can reflect the same dynamics described by the Hamiltonian in (1).

If considering the influence of environment, the total Hamiltonian can be supposed as , where 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

where is a projected state, , and is a similarity operator. These parameters and relations can be constructed by the following several rules.

(1) Propose projectors as

and , where corresponds to the index of eigenvectors for Hamiltonian in system, denotes the index to the Hamiltonian in environment, and is the th eigenvector of diagonal part of the interaction Hamiltonian between the system and environment.

(2) Introduce the creation (destruction) operator expressed as

(3) Introduce the projected state in the subdynamic system as

where is a solution of the original Schrödinger equation and the similarity operator is defined by

(4) The ruling equation is subdynamic Liouville equation:

where operator is introduced by

and the relationship of the above operator with the original can be proven by a kind of similarity transformation:

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 between (or ) and (or ) 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, , the Liouville equation for this two-level system with the Rabi oscillation is expressed by

where one notices that using (13) one has

Then, from (13) one gets the time-independent equations of as

Then by introducing , , as a Bloch vector, that is,

with

where , , is an unit vector, respectively, and , thus (15) can be changed as

Then using the subdynamic similarity operator above, one gets

The solution of (20) is given by

where

so (20) can also give a solution based on the Bloch ball,

where 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 and else three states, , , and as zero), one gets

This allows one, from (23), to get

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

where or expressed the same meaning as in (1) and is a coupling number reflecting dissipation velocity. So the corresponding master equation for the pseudovector is expressed as

then through the subdynamic similarity operator one obtains

where we notice that ; , expresses the life time of state , and is a coupling number reflecting pure phase decoherence. Then considering (18) one can define

This allows us to introduce the subdynamics for or by the above formalism:

with

by defining the projector

with

so that is expressed by the subdynamic formalism:

This can be measured in the practical system with energy term , and then using , we can construct the projected states in the subdynamic space by the subdynamic formalism:

where it is marvelous that whatever the system has or , the above formalism gives the same in the subdynamic space; namely, if of the computing system changes to then the similarity operator in (22) changes to

which, from the changed , still gives the same as

Hence this equation group allows one, from the measurement of states and energy , for , to determine the projected states . These projected states are invariant in the subdynamic space in the sense:

in which both cases have the same eigenvectors , although the eigenvalues are changed from to . So from (21) one can obtain

where one notices that the evolution is given by

Hence based on the above subdynamic formalism the physical realizing states in the subdynamic space are related to which can be determined by the states .

The remaining change of the eigenvalue from to , 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 ; then the phase error correction in the evolution of is only necessarily considered. This can be given by using (39):

so the correction is to replace by and replace by , which enables (41) to return (21). For example, using the quantum three-editing-code method [23] if there are phase errors in for a quantum computing tolerance time then one can correct the errors by using

where means because has a phase error. Thus by means of one can conclude which qubit has error. This process is described by an error point formula: . Then the correction is to replace by through replacing the relevant eigenvalues after considering change of the eigenvalues. This process can keep the working states to be corrected by canceling the time evolution phase error of the states during every error tolerance time interval 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, where is vacuum state of excitons and and are or polarization of exciton states, respectively, while is states of two excitons. Thus using and polarization light, the CROT for two qubits can be realized as

with

which gives an operation on the states based on (17) as

so that changes to . Therefore in the subdynamic space one can perform the CROT operation for two-qubit system ideally; that is,

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 of 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the support of this work from Wuhan University of Technology.

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