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By removing a ^{12}C atom from the tetrahedral configuration of the diamond, replacing it by a ^{13}C atom, and repeating this in a linear direction, it is possible to have a linear chain of nuclear spins one half and to build a solid state quantum computer. One qubit rotation, controlled-not (CNOT) and controlled-controlled-not (CCNOT) quantum gates are obtained immediately from this configuration. CNOT and CCNOT quantum gates are used to determined the design parameters of this quantum computer.

So far, the idea of having a working quantum computer with enough number of qubits (at least 1000) has faced two main problems: the decoherence [1-8] due the interaction of the environment with the quantum system, and technological limitations (pick up signal from NMR quantum computer [9,10], laser control capability in ion trap quantum computer [11,12], physical build up for more than two qubits like in photons cavities [^{12}C (with nuclear spin zero) configuration of the diamond main structure, one removes a ^{12}C element of this configuration and replace it by a ^{13}C (with nuclear spin one half) atom, and one repeats this replacement along a linear direction of the crystal. By doing this replacement, one obtains a linear chain of atoms of nuclear spin one half which is protected from the environment by the crystal structure and the electrons cloud. Therefore, one could have a quantum computer highly tolerant to environment interaction and maybe not so difficult to build it, from the technological point of view.

The above idea is represented in ^{13}C atoms are place on the position of some ^{12}C atoms. This replacement could be done using the same technics used to construct the diamond NV structure [^{13}C in the diamond [

Now, as one can see, the important interaction on this configuration is the spin-spin interaction between the nucleus of the ^{13}C atoms. This interaction is well known [

where the magnetic moment ^{13}C’s is related with the nuclear spin as

being ^{13}C magnetic moment is due to proton. The variable ^{13}C nucleus, which has magnitude^{13}C nucleus along the x-axis of the reference system and assuming Ising interaction between ^{13}C nucleus, this energy can be written as

where the coupling constant

Consider a magnetic field of the form

where^{13}C’s nuclear magnetic moments,

The magnetic field at the location of the ith-^{13}C atom is^{13}C atoms with the magnetic field is

where ^{13}C atoms aligned along the x-axis. This energy can be written as

where ^{13}C,

Let us consider first and second neighbor interactions among ^{13}C nuclear spins, and assuming equidistant separation between any pair of spins, the Hamiltonian of the system is

where ^{13}C atoms, and ^{13}C atoms which must be about one order of magnitude lower than

and

The operator

Since

The Schrödinger’s equation,

is solved by proposing a solution of the form

which brings about the following system of first order differential equations on the interaction representation

where

and

This is very well known procedure to solve time dependent Schrödinger’s equation, and the solution of Equation (19) brings about he unitary evolution of the system (given the initial condition

Defining the evolution parameter

In order to get an operating quantum computer, one needs to show that, at least, one qubit rotation gate ^{12}C-^{13}C diamond system. However, one needs to assign realistic workable parameters for the real design of a ^{12}C-^{13}C diamond quantum computer. To do this, one studies in this section the behavior of a quantum CNOT and CCNOT gates as a function of several parameters. One neglect one qubit rotation ^{13}C atom in the diamond structure. In particular, the NOT quantum gate is obtained using a

Larmore’s frequencies are denoted by

where ^{13}C nucleus,

For the CNOT quantum gate, one has the initial conditions

at the end of the

^{13}C atoms about the the length of the diamond unit cell, one can select

One needs to mention that in the case the alignment of the ^{13}C atoms be along the z-axis (the same direction of

the longitudinal magnetic field), the coupling constant deduced from Equation (1) would be given by

According to these results, one has now an idea of the value of the parameters for the design of a quantum computer with the ^{12}C - ^{13}C diamond quantum system: (a) Separation between ^{13}C atoms is

For the CCNOT quantum gate, one has the initial conditions

Although the gradient of the magnetic field might be a concern, the magnitude of the longitudinal magnetic field is low enough to think that this gradient can be achieved. The scalability of the system is clear, the read out system could be based on single spin measurement technics [

It was shown that by removing a ^{12}C atom, replacing it by a ^{13}C atom in the tetrahedral configuration of the diamond, and doing this process periodically in a linear direction, one could get a linear chain of nuclear spins one half which can work as a quantum computer. The interaction between ^{13}C atoms is governed by the magnetic dipole-dipole interaction, and the parameters of a possible quantum computer design were determined by studying the quantum CNOT and CCNOT gates with two and three qubits respectively. Although there might be a concern about the gradient of the magnetic field along the lines of ^{13}C atoms, it must not be so difficult to get this gradient since the magnitude of this magnetic field is relatively low (0.5 T). In principle, it is possible to replace a ^{12}C atom by any other spin one half atom. However, an unclose configuration of electrons in the lattice makes necessarily to take into account the interaction of electrons with this atom (as it is the case of diamond NV configuration) which makes the analysis and the quantum computer much more complicated and sensitive to environment interaction. The misplacement of the ^{13}C atom along the x-axis produces different coupling constant in the interaction, but according to ^{13}C atoms off x-axis changes the coupling constant and the interaction itself, which has to be studied. In addition, it still remains to study the decoherence on this system. Finally, one recalls that the carbon isotopesatoms ^{12}C, ^{13}C and ^{14}C occur naturally on Earth with a percent of about 99%, 1% and 10^{−4}%, ^{14}C being a radioactive isotope with a half life of about 5730 years and having spin zero. Therefore,^{ 14}C is not an important composition at all in the diamond structure.

Two qubits dynamics is obtained from Equations (13), (14), and (19), resulting the equations

where the complex variables

As before, three qubits dynamics is described by the equations

with

For both cases, the energies