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Advances in Mathematical Physics
Volume 2010 (2010), Article ID 365653, 10 pages
http://dx.doi.org/10.1155/2010/365653
Research Article

Quantum Computing in Decoherence-Free Subspace Constructed by Triangulation

1Department of Physics, School of Science, Wuhan University of Technology, Wuhan 430070, China
2Centre for Advanced Nanotechnology, University of Toronto, Toronto, M5S 3E4, Canada

Received 17 August 2009; Revised 2 November 2009; Accepted 30 December 2009

Academic Editor: Nai-Huan Jing

Copyright © 2010 Qiao Bi et al. 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 formalism for quantum computing in decoherence-free subspaces is presented. The constructed subspaces are partial triangulated to an index related to environment. The quantum states in the subspaces are just projected states which are ruled by a subdynamic kinetic equation. These projected states can be used to perform ideal quantum logical operations without decoherence.

1. Introduction

Recent publications have formulated a remarkable theory for Decoherence-Free (DF) subspaces and subsystems in which quantum computing is performed in a DF subspace although the total space is still subject to decoherence [123]. Many proposals have considered the Born-Markov approximation or restrictions on the type of decoherence (e.g., symmetric and collective decoherence). However, there have been a number of reports on DF subspaces that do not invoke the Born-Markov approximation and do not place constraints on the type of decoherence [24]. These have significance for practical experimental implementation. So far, the most general framework may be called operator quantum error correction (OQEC), which encompasses active error correction and leads to improved threshold results in fault tolerant quantum computing [2528], and has initiated the development of a structure theory for passive error correction [29, 30]. Experimentally, DF subspaces have recently been observed under some conditions, which shows that such DF subspaces do indeed exist, allowing logical qubits to be encoded without decoherence [2234]. However, a practical procedure to construct DF subspace is still not a trivial problem, and hence, impedes building practical quantum logic gates. For this study, we present a different scheme to construct a DF subspace based on the Schrödinger type of subdynamic kinetic equation (SSKE) which was inspired by theory of the subdynamics in Brussels-Austin school [35, 36].

2. Subdynamic Formalism

Consider a general quantum open system 𝑆 interacting with an environment 𝐵, whose Hamiltonian is written as 𝐻(𝑡)=𝐻𝑆(𝑡)+𝐻𝐵+𝜆𝐻int with coupling number 𝜆. If one chooses the eigenprojectors and its orthogonal projectors of free Hamiltonian 𝐻0(𝑡)=𝐻𝑆(𝑡)+𝐻𝐵 as 𝑃𝑛 and 𝑄𝑛, respectively (this is possible since the free Hamiltonian is assumed to be easily diagonalized), then by means of the subdynamics theory [37] one can introduce a creation (destruction) correlation operator (as a type of resolvent) as 𝐶𝑛1(𝑡)=𝐸𝑛(𝑡)𝑄𝑛𝐻(𝑡)𝑄𝑛𝑄𝑛𝐻1(𝑡)𝑃𝑛=𝐷𝑛(𝑡),(2.1) where 𝐸𝑛(𝑡) is an eigenvalue of 𝐻(𝑡). This allows one to construct a Schrödinger type of kinetic equation for a projected state as 𝑖𝜕𝜙𝜕𝑡proj(𝑡)=Θ(𝑡)𝜙proj(𝑡),(2.2) where the projected wave function 𝜙proj is defined as 𝜙proj(𝑡)=𝑛𝑃𝑛Π𝑛(𝑡)𝜑(𝑡),(2.3) where the operator Π𝑛(𝑡) is expressed by Π𝑛𝑃(𝑡)=𝑛+𝐶𝑛𝑃(𝑡)𝑛+𝐷𝑛(𝑡)𝐶𝑛(𝑡)1𝑃𝑛+𝐷𝑛(𝑡),(2.4) and the wave function 𝜑(𝑡) satisfies the original Schrödinger equation, while the intermediate operator Θ is defined as Θ(𝑡)=𝐻0(𝑡)+𝜆𝑛𝑃𝑛𝐻1(𝑡)𝐶𝑛(𝑡).(2.5) The creation operator 𝐶𝑛(𝑡) is independent of the representation with respect to the projectors 𝑃𝑛, 𝑄𝑛 and is not necessarily self-adjoint in the sense of extended functional space. This reveals that the eigenvalues of the intermediate operator may be complex and the corresponding evolution operator corresponding to two nonunitary semigroups evolutions, respectively. Using the physical boundary conditions, such as 𝜖-rule [35], we can determine which semigroup is the correct one. Therefore, the evolution of the projected density operator for the open system can be time asymmetric and may exist in the generalized functional space beyond Hilbert space, such as rigged Hilbert space [38, 39]. This time asymmetric evolution is consistent with the second law of thermodynamics.

Moreover, by replacing the Hamiltonian 𝐻(𝑡) with the Liouvillian 𝐿(𝑡) and wave function 𝜑 with density operator 𝜌, as well as using the same approach as above, the SSKE can be transformed to the Liouvillian type of SKE (LSKE) for a quantum open system: 𝑖𝜕𝜌𝜕𝑡proj(𝑡)=Θ(𝑡),𝜌proj(𝑡).(2.6)

The creation (destruction) correlation operator can be available by means of subdynamics theory. In fact, by introducing the eigen-projectors of the total Hamiltonian Π𝑛(𝑡), using the Heisenberg equation considering the eigenvalue problem of Π𝑛(𝑡) and the definition of the creation (destruction) operators, one can obtain the basic operator equation of the creation operator and its solutions can be expressed as the retarded or advanced integrals, corresponding to the two kinds of time evolution semigroups: 𝑡[0,+) or 𝑡(,0] and continuations of up and down half complex planes. In the same way, the basic equation for the destruction operator and relevant solution can be given.

This construction of SSKE or LSKE in subspace can be related to the original Schrödinger or Liouville equation. For instance, using the intertwining relations from subdynamics theory [35, 36, 40] Ω(𝑡)Θ(𝑡)Ω1(𝑡)=𝐻(𝑡), one can arrive at the original Schrödinger equation 𝑖𝜕𝜕𝑡𝜙(𝑡)=Ω(𝑡)Θ(𝑡)Ω1(𝑡)𝜙(𝑡)=𝐻(𝑡)𝜙(𝑡),(2.7) where the similarity operator is defined as Ω(𝑡)=𝑛𝑃𝑛+𝐶𝑛(𝑡),(2.8) which is also not necessarily unitary. The eigenvectors of the time-independent total Hamiltonian 𝐻 can be given by the eigenvectors of 𝐻0, 𝜙𝑛, as |𝜑𝑛=(𝑃𝑛+𝐶𝑛)|𝜙𝑛 with the same structure of eigenvalues as Θ.

The second-order approximation for the LSKE also corresponds to the Master, Boltzmann, Pauli, and Fokker-Plank equations of kinetic theory and Brownian motion. For example, from the LSKE for a time-independent open system 𝑆, one can easily deduce the general Master equation by using Born-Markovian approximation. Indeed, if assuming that projector as 𝑃=exp𝛽𝐻𝐵𝑇𝑟𝐵exp𝛽𝐻𝐵𝑇𝑟𝐵(2.9) giving the reduced density operator for system 𝑆 as 𝑃𝜌=𝜌𝑆, then a general Markovian equation can be obtained. This equation can also be reduced to a type of Lindblad equation or Boltzmann equation [41].

3. Decoherence-Free Partial Triangular Subspace

An interesting advantage using the above formalism is to construct a precise decoherence-free (DF) subspace. It is remarkable that the projected space on which subdynamics operates is a kind of DF subspace that occurs naturally by choosing a suitable basis to expand that subspace. In fact, if decoherence exists in a system arising from interactions with its environment, then the spectral decomposition of the Hamiltonian can be expressed using the subdynamic formalism as 𝐻0=Θ0=𝑛𝐸0𝑛(𝑡)𝑃𝑛𝑛𝐸0𝑛(𝑡)+𝐸1𝑛𝑃𝑛,(3.1) where it is not very difficult to see that the eigenprojector 𝑃𝑛 for the free Hamiltonian 𝐻0(𝑡) is invariant while the eigenvalue 𝐸0𝑛(𝑡) is changed to 𝐸0𝑛(𝑡)+𝐸1𝑛. This produces a phase shift for the evolution state 𝜙proj,𝑛𝐸(𝑡)=exp𝑖0𝑛(𝑡)+𝐸1𝑛𝑡𝜙proj,𝑛(0),(3.2) which leads to a type of decoherence in the subspaces. For example, if the entangled states evolution in the subspace is 𝑛exp(𝑖𝐸0𝑛(𝑡)𝑡)𝜙𝑛(0) before the interaction is exerted from the environment, then after the interaction the evolution of the state in the subspace becomes 𝑛exp𝑖𝐸0𝑛(𝜙𝑡)𝑡𝑛(0)𝑛𝐸exp𝑖0𝑛(𝑡)+𝐸1𝑛𝑡𝜙𝑛(0).(3.3) Hence for constructing an ideal DF subspace, one has to find a procedure to cancel the change of the eigenvalues. How can one realize it? The key idea is to use the partial triangulation.

For example, let us consider a typical two-qubit quantum computing system 𝑆, consisting of the spins 𝐒1 and 𝐒2, such as the two electrons of two 31𝑃 confined in a germanium/silicon heterostructure of an electron spin-resonance transistor [42, 43] or the two electrons confined in two quantum dots [44]. Ignoring the influence of the environment, the Hamiltonian can be written using the Heisenberg model as 𝐻𝑆(𝑡)=𝐽(𝑡)𝐒1𝐒2, where 𝐽(𝑡) is the time-dependent exchange coupling parameter determined by the specific model considerations. In the case of spins of the two electrons (e.g., confined in two vertically, laterally, coupled quantum dots [45]), 𝐽 is the difference in the energies of two-electrons ground state, a spin singlet at zero magnetic field, and the lowest spin-triplet state; 𝐽 is also a function of the electric and magnetic field and the interdot distance. Using the relationship between 𝐒1𝐒2 and the square of the sum of 𝐒1 and 𝐒2, the eigenvalues and eigenvectors of 𝐒1𝐒2 can be found from 𝐒1𝐒2=(1/2)(𝐒2(3/2)) by 𝐸1=12||𝜙1||||𝜙=11,2||||𝜙=00,31=2||||,𝐸01+1023=4||𝜙41=2||||0110.(3.4) A quantum Controlled-Not (CN) gate can be given by the sequence of operations 𝑈CN=𝑒𝑖(𝜋/2)𝑆𝑧1𝑒𝑖(𝜋/2)𝑆𝑧2𝑈1/2sw𝑒𝑖𝜋𝑆𝑧1𝑈1/2sw,(3.5) where 𝑈sw is an ideal swap operator which can exchange the quantum states of qubits 1 and 2 and is determined generally by an evolution operator 𝑈𝑠(𝐽(𝜏)) by adjusting the coupling time between the two spins in the evolution of the system. For the particular spin-spin coupling duration 𝜏𝑠, where 𝜏𝑠0𝐽(𝑡)𝑑𝑡=𝜋  (mod2𝜋), 𝑈sw=𝑈𝑠(𝜋), and the swap operator which can exchange the quantum states of qubits 1 and 2 is given by [44] 𝑈sw=𝑒𝑖𝜏𝑠0(𝐻𝑆(𝜏))𝑑𝜏=3𝑛=1𝑒𝑖(𝜋/4)||𝜙𝑛𝜙𝑛||+𝑒𝑖(3𝜋/4)||𝜙4𝜙4||.(3.6)

In the presence of the environment, the nonideal action of the swap operation must be considered because it may introduce decoherence in the ideal swap operation. Here the environment is assumed to consist of a set of two-level particles randomly embedded in an environment 𝐵. This is a pure dephasing model whose Hamiltonian is given by 𝐻𝐵=𝑘𝜔𝑘𝜎𝑧𝑘, and the Hamiltonian coupling the two-qubit spin system is 𝜆𝐻int=𝑘𝜎𝑧1+𝜎𝑧2𝑔𝑘𝜎+𝑘+𝑔𝑘𝜎𝑘,(3.7) where 𝜎+𝑘 (𝜎𝑘) is raising/lowering operator for the 𝑘th two-level particle, characterized by a generally complex coupling parameter 𝑔𝑘. Then, the Hamiltonian for the total system is 𝐻(𝑡)=𝐻𝑆(𝑡)+𝐻𝐵+𝜆𝐻int, and the corresponding complete set of eigenvectors for 𝐻𝑆(𝑡)+𝐻𝐵 is denoted as {|{𝑘}𝜙𝑗,𝜙𝑗{𝑘}|}.

To control the induced decoherence, we choose the time-independent eigen-projectors of 𝐻𝑆(𝑡)+𝐻𝐵 as 𝑃𝑛𝑘||𝜑𝑛𝑘𝜑𝑛𝑘||=||𝜙𝑛𝜙𝑛||||||{𝑘}{𝑘}(3.8) and 𝑄𝑛𝑘 as 𝑄𝑛𝑘+𝑃𝑛𝑘=1, 𝑛=1,,4. Using the definition of the eigen-projectors 𝑃𝑛𝑘, the spectral decomposition of the intermediate operator Θ is given by two cases: Θ(𝑡)=4𝑛=1𝜑𝑛𝑘||𝐻𝑆(𝑡)+𝐻𝐵||𝜑𝑛𝑘𝑃𝑛𝑘,withouttheinteraction,(3.9)Θ(𝑡)=4𝑛=1𝜑𝑛𝑘||𝐻𝑆(𝑡)+𝐻𝐵||𝜑𝑛𝑘𝑃𝑛𝑘+𝜑1𝑘||𝜆𝐻int𝑄1𝑘1𝐸1𝑘(𝑡)𝑄1𝑘𝐻(𝑡)𝑄1𝑘𝑄1𝑘𝜆𝐻int||𝜑1𝑘𝑃1𝑘+𝜑2𝑘||𝜆𝐻int𝑄2𝑘1𝐸2𝑘(𝑡)𝑄2𝑘𝐻(𝑡)𝑄2𝑘𝑄2𝑘𝜆𝐻int||𝜑2𝑘𝑃2𝑘,withtheinteraction,(3.10) where 𝐸1𝑘(𝑡) and 𝐸2𝑘(𝑡) are the eigenvalues of the total Hamiltonian, corresponding to the eigen-projectors of Θ(𝑡), 𝑃1𝑘, and 𝑃2𝑘, respectively. This shows that the eigen-projectors 𝑃𝑛𝑘 of Θ(𝑡) are invariant and independent of the interaction in the constructed subspace; however, for 𝑛=1,2, the eigenvalues are changed (index 𝑛 is still diagonal to 𝐻int, while index 𝑘 is off-diagonal to 𝐻int), which may introduce a phase shift in the evolution operator as a kind of decoherence.

For canceling this phase shift, we consider the triangular decomposition of the 𝐻int as 𝐻utriint+𝐻dtriint, where 𝐻utriint is upper-triangular part of 𝐻int and 𝐻dtriint is lower-triangular part of 𝐻int. Then it is easy to find that 𝜑𝑛𝑘||𝐻utriint+𝐻dtriint1𝐸𝑛𝑘(𝑡)𝑄𝑛𝑘𝐻(𝑡)𝑄𝑛𝑘𝐻utriint+𝐻dtriint||𝜑𝑛𝑘𝑃𝑛𝑘=𝜑𝑛𝑘||𝐻utriint1𝐸𝑛𝑘(𝑡)𝑄𝑛𝑘𝐻(𝑡)𝑄𝑛𝑘𝐻dtriint||𝜑𝑛𝑘𝑃𝑛𝑘+𝜑𝑛𝑘||𝐻dtriint1𝐸𝑛𝑘(𝑡)𝑄𝑛𝑘𝐻(𝑡)𝑄𝑛𝑘𝐻utriint||𝜑𝑛𝑘𝑃𝑛𝑘,(3.11) which shows that 𝐻utriintpart and 𝐻dtriint part are to be moved out in the upper-triangular or lower-triangular subspace Φutri(Φdtri), respectively. Here the upper-triangular (or lower-triangular) subspace Φutri(Φdtri) can be defined by introducing upper-triangular (or lower-triangular) projector and upper-triangular (or lower-triangular) inner product given by 𝑃𝑘𝑘=|𝑘𝑘|, 𝑘𝑘(or𝑘𝑘), and 𝑘|𝐴|𝑘, for any operator 𝐴 with 𝑘𝑘(or𝑘𝑘). For instance, any operator 𝐴 defined in this upper-triangular subspace with respect to the index 𝑘 can only be represented as 𝐴=𝑘𝑘𝑘|𝐴|𝑘|𝑘𝑘|. Therefore we can construct the DF upper-triangular (or lower-triangular) subspace Φutri with respect to the environmental index 𝑘 and, in the same time, enable the Hilbert space 𝑆 to be invariant for the quantum computing system 𝑆, that is, 𝑆Φutri, in which one can allow the interaction terms in (3.10) to be zero 𝜑𝑗𝑘||𝜆𝐻int𝑄𝑗𝑘1𝐸𝑗𝑘(𝑡)𝑄𝑗𝑘𝐻(𝑡)𝑄𝑗𝑘𝑄𝑗𝑘𝜆𝐻int||𝜑𝑗𝑘𝑃𝑗𝑘=0,for𝑗=1,2.(3.12) This demonstrates that the constructed intermediate operator Θ(𝑡) on the upper-triangular subspace 𝑆Φutri is independent upon the interaction part of the original Hamiltonian 𝐻int; consequently the phase shift introduced by 𝐻int is canceled in the subspaces although the total system experiences the decoherence introduced by 𝐻int.

In the upper-triangular subspace 𝑆Φutri, the quantum Controlled-Not logic operation 𝑒𝑖(𝜋/2)𝑆𝑧1𝑒𝑖(𝜋/2)𝑆𝑧2𝑈1/2sw𝑒𝑖𝜋𝑆𝑧1𝑈1/2sw can be executed by using a sequence of operations and its relevant swap operator 𝑈sw remains invariant before and after interaction from the environment. This can be described by the following formula: 𝑇𝑟𝐵𝑒𝑖𝜏𝑠0Θ(𝜏)𝑑𝜏=𝑇𝑟𝐵𝑒𝑖𝜏𝑠0(𝐻𝑆(𝜏)+𝐻𝐵)𝑑𝜏=3𝑛=1𝑒𝑖(𝜋/4)||𝜙𝑛𝜙𝑛||+𝑒𝑖(3𝜋/4)||𝜙4𝜙4||×𝑘𝑒𝑖(𝑎2𝜔𝑘+𝜔𝑘+1)/(𝑎2+1)+𝑒𝑖(𝑎2𝜔𝑘+1+𝜔𝑘)/(𝑎2+1).(3.13) The above formalism can be extended to a general dephasing situation, for instance, a register of 𝑁 two-level elements immersed in a quantized environment, whose Hamiltonian is described by 𝐻=𝑁1𝑛=0𝜀𝜎𝑧(𝑛)+𝑘𝜔𝑘𝑏𝑘𝑏𝑘+𝑁1𝑛=0𝑘𝜎𝑧(𝑛)𝜒𝑘𝑛𝑏𝑘+𝜒𝑘𝑛𝑏𝑘.(3.14) From the preceding study, the procedure to construct the upper-triangular subspaces is simple, that is, if one performs the relevant quantum logical operations using the state 𝜙(0), then one should perform the normal inner product with respect to the index 𝑛 for the quantum computing system and the upper-triangular inner product with respect to the index 𝑘 for the environment, and remember that the evolution of the states is described by exp(𝑖Θ(𝑡)𝑡)𝜙(0) in the upper-triangular subspace. The general considerations for the above model are given below.

4. General DF Subspace Constructed by Triangulation

Different models treat the interaction of the system with its environments quite differently; here, we propose a general procedure to construct a DF subspace by triangulation. Suppose that the states used in quantum computing system are the eigenvectors of the free Hamiltonian 𝐻0=𝐻𝑆(𝑡)+𝐻𝐵, then the corresponding matrix of the Hamiltonian 𝐻0 is diagonal. Thus the spectral decomposition for the intermediate operator Θ based on the above subdynamics formalism can be given by Θ=𝑇𝑟𝑘(𝑛,𝑘𝐸𝑛(𝑡)𝑃𝑛𝑘)=𝐻𝑆(𝑡), where 𝑃𝑛𝑘 is an eigen-projector of 𝐻0, and 𝐸𝑛(𝑡) is an eigenvalue of 𝐻𝑆.

Now, if the system is subject to decoherence induced from the environment by (a general) 𝐻int, then (suppose) the matrix of Hamiltonian becomes off-diagonal to the index introduced from environment (and is diagonal to the index from the original system). Thus, one can construct the upper-triangular subspace 𝑠Φutri by defining the rule of upper-triangular inner product in this space such that 𝜑𝑛𝑘|𝐻int|𝜑𝑛𝑘0, for 𝑘𝑘; 𝜑𝑛𝑘|𝐻int|𝜑𝑛𝑘=0, only for the index related to the environment 𝑘>𝑘. This leads to an upper-triangular matrix of 𝐻int. Using this upper-triangular property, the interaction terms in the Θ operator should be zero 𝑘𝑘𝑃𝑛𝑘𝐻int𝑃𝑛𝑘𝐶𝑛𝑘𝑃𝑛𝑘=0; thus, the spectral decomposition of Θ=𝐻𝑆 maintains invariance, that is, Θ=𝑇𝑟𝑘(𝑛,𝑘𝐸𝑛(𝑡)𝑃𝑛𝑘+0)=𝐻𝑆, where 𝑃𝑛𝑘 is now chosen as an upper-triangular eigen-projector with respect to 𝑘.

It may be necessary to emphasize that the states required to perform quantum computing in the DF (triangular) subspace are just the projected states 𝜓proj(𝑡) which can be spanned by {𝜑𝑛𝑘=𝜙𝑛𝜓𝑘}. The projected states are measurable in this subspace since {𝜑𝑛𝑘=𝜙𝑛𝜓𝑘} is orthogonal (and triangular with 𝑘) and distinguishable, where the triangular inner product only is valid to the index which is related to the environment; here it means 𝑘, while for an other index relevant to the original system, such as 𝑛, the normal inner product is still used. Finally, one may ask whether the rule of the upper-triangular inner product in this space restricts or changes the original Controlled-Not logic operation and results in errors. The answer is no, as one can see there is no influence on decoherence to the quantum Controlled-Not logic operation in (3.13). The term relative to the environment 𝑘𝑒𝑖(𝑎2𝜔𝑘+𝜔𝑘+1)/(𝑎2+1)+𝑒𝑖((𝑎2𝜔𝑘+1+𝜔𝑘)/(𝑎2+1))(4.1)

now cannot influence the original quantum Controlled-Not logic operation 3𝑛=1𝑒𝑖(𝜋/4)||𝜙𝑛𝜙𝑛||+𝑒𝑖(3𝜋/4)||𝜙4𝜙4||.(4.2)

The entanglement between the system and environment is canceled in the constructed partial triangular subspace although it exists indeed in the original total space or even in the subspace with the normal inner product. The role of upper-triangular inner product with respect to the index from environment in the subspace is only to cancel the decoherence (phase shift) from the environment. Then, how can one realize the above procedure in the practical procedure for the quantum computing? We suppose to establish an additional measure or count system to read or calculate some dates, such as the eigenvalues and eigenvectors from the original system, and transfer the relevant dates to the expression in the frame of the partial triangular subspace based on (2.2) and the rule of the partial triangular product. Here the key is to establish a transformation system for the constructed partial triangular subspace, which allows the eigenvalues and eigenvectors from the original system to be expressed in the partial triangular subspace.

5. Conclusions and Remarks

A scheme for quantum computing in the DF triangular subspaces is presented. The DF subspaces are ruled by the subdynamic kinetic equation (SKE). The used quantum computing states in the DF subspaces are just the projected states 𝜙proj(𝑡) in the DF triangular subspaces. Moreover, this DF subspace is partial upper-triangular, in which an inner product is only upper-triangulated to index 𝑘 corresponding to environment. That means, in a quantum computing process, that if one takes the upper-triangular inner product with respect to 𝑘 and keeps the ordinary inner product to be invariance with respect to the index related to original quantum computing system, then the decoherence in the subdynamic spaces can be completely cancelled. The Markovian and non-Markovian decoherence and collective decoherence as studied in recent publications seem not to present any restrictions on the DF subspace considered here, and this proposal may be useful generally for the register of 𝑁 two-level elements immersed in a quantized environment.

Acknowledgment

This work was supported by the grants from the National Natural Science Foundation of China (Grant no. 60874087) as well as Canadian NSERC, CIPI, MMO, and CITO.

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