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 [13], atoms traps [14,15], Josephson’s joint ions [16], Aronov-Bhom devices [17], diamond NV device [18], or high field and high field gradients in linear chain of paramagnetic atoms with spin one half [19]). In particular, the linear chain of paramagnetic atoms of spin one half became a good mathematical model to make studies of quantum gates [20], quantum algorithms [21], and decoherence [22] which could be applied to other quantum computers. In this paper, one put together the ideas of using the diamond stable structure and the linear chain of spin one half nucleus. To do this, on the tetrahedral ¹²C (with nuclear spin zero) configuration of the diamond main structure, one removes a ¹²C element of this configuration and replace it by a ¹³C (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 Figure 1, where the ¹³C atoms are place on the position of some ¹²C atoms. This replacement could be done using the same technics used to construct the diamond NV structure [23], or using ion implantation technics [24] and neutralization of ¹³C in the diamond [25]. It is assumed in this paper that this configuration can be built somehow.

Now, as one can see, the important interaction on this configuration is the spin-spin interaction between the nucleus of the ¹³C atoms. This interaction is well known [26] and is given by

where the magnetic moment of ¹³C’s is related with the nuclear spin as

being the proton gyromagnetic ratio . Without loosing the main idea, it will be assumed here that ¹³C magnetic moment is due to proton. The variable indicates the separation vector between two ¹³C nucleus, which has magnitude. Aligning the chain of ¹³C nucleus along the x-axis of the reference system and assuming Ising interaction between ¹³C nucleus, this energy can be written as

where the coupling constant has been defined as

Consider a magnetic field of the form

where, , and are the magnitude, the phase, and the frequency of the transverse rf-field. The z-component of the magnetic field has a gradient on the x-axis, determined by the difference on Larmore’s frequencies of the ¹³C’s nuclear magnetic moments,

The magnetic field at the location of the ith-¹³C atom is, and the interaction energy of the magnetic moments of the ¹³C atoms with the magnetic field is

where is the number of ¹³C atoms aligned along the x-axis. This energy can be written as

where is the Larmore’s frequency of the ith-¹³C,

is the Rabi’s frequency,

and are the ascent and descent spin operators, , and has been defined as

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

where is the coupling constant of first neighbor ¹³C atoms, and is the coupling constant of second neighbor ¹³C atoms which must be about one order of magnitude lower than. One can write this Hamiltonian as, where and are defined as

and

The operator is diagonal on the basis of the Hilbert space of dimensionality. Its eigenvalues defines the spectrum of the system,

Since for, this spectrum is not degenerated with as the energy of ground state, and as the energy of the most exited state. To calculate the spectrum, one has used the following action of operator

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 are defined as

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 through the change of variable , the parameters, , and are real numbers given in units of. This evolution parameter will be used below in the analysis of the CNOT quantum gate.

In order to get an operating quantum computer, one needs to show that, at least, one qubit rotation gate and two qubits CNOT gate or three qubits controlled-controlled-not (CCNOT) gate can be constructed from this quantum system. Because this quantum system is homeomorphic [27] to the linear chain of paramagnetic atoms with spin one half system [28], it is clear from the point of view of mathematical models that the above gates can be constructed with this ¹²C-¹³C diamond system. However, one needs to assign realistic workable parameters for the real design of a ¹²C-¹³C 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 because it is obvious that one can get it through an arbitrary pulse on the rf-field with the frequency given by the Larmore's frequency of the qubit, for a single ¹³C atom in the diamond structure. In particular, the NOT quantum gate is obtained using a -pulse duration with this frequency. Therefore, the study of the CNOT and CCNOT quantum gates is of the most interest. The equations for the two and three qubits dynamics are shown on the appendix. CNOT quantum gate corresponds to the transition, and CCNOT quantum gate corresponds to the transition. The first and second transitions are obtained through the resonant frequencies

Larmore’s frequencies are denoted by and, and is parametrized as

where measures the relative change of the frequencies of qubits. The separation of the ¹³C nucleus, , is parametrized as

For the CNOT quantum gate, one has the initial conditions and . The time is allowed to last a -pulse, and one takes the coupling constant as a fixed paramenter,

Figure 2 shows the fidelity parameter,

at the end of the -pulse, as a function of the Rabi's frequency, where is the state obtained with the simulation, and is the expected state. The simulation was done for two different weak magnetic fields and for (1), (2), (3), and (4). The oscillations seen on this picture are due to the low and high contribution of the non resonant states (and) to the dynamics of the system, which depends on Rabi’s frequency and they are explained by the -method [19]. As one can see from this picture , the CNOT gate is very well produced either with and or with and.

Figure 3 shows the gradient of magnetic field along the x-axis, the coupling constant, and the fidelity of the CNOT quantum gate as a function of the two qubits separation (characterized by the parameter, Equation (24)), having. As one can see, the fidelity is not sensitive for relatively wide variation of, meanwhile the gradient and coupling constant have the strong variation deduce from Equation (6) and Equation (4). Considering the separation of the two ¹³C atoms about the the length of the diamond unit cell, one can select, corresponding to a coupling constant of, and a magnetic field gradient of.

One needs to mention that in the case the alignment of the ¹³C 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, with given by Equation (4), and basically the results are the same as the presented here.

According to these results, one has now an idea of the value of the parameters for the design of a quantum computer with the ¹²C - ¹³C diamond quantum system: (a) Separation between ¹³C atoms is which can be aligned along the x-axis, (b) coupling constant is, (c) longitudinal magnetic field is, (d) gradient of this longitudinal magnetic field along the x-axis is , and (e) magnitude of the rf-magnetic field on the plane x-y is (Rabi's frequency).

For the CCNOT quantum gate, one has the initial conditions for and. The time is allowed to last a -pulse. The coupling parameter is taken as before, and second neighbor coupling parameter. Figure 4 shows the fidelity parameter as a function of Rabi’s frequency for magnetic field intensities (1), (2)

, (3), and (4), for (a), and for (b). As one can see fro these plots, a good CCNOT quantum gate can be obtained by choosing (implying and encresing of the gradient of the magnetic field by a factor of two), and with the other parameters given as defined with the CNOT quantum gate.

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 [29], and studies on decoherence remains to be done on this system. This quantum computer resembles a solid state NMR system [30].

It was shown that by removing a ¹²C atom, replacing it by a ¹³C 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 ¹³C 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 ¹³C 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 ¹²C 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 ¹³C atom along the x-axis produces different coupling constant in the interaction, but according to Figure 4, the fidelity of the CNOT quantum gate does not change, and one would expect the same result for quantum algorithms. The displacement of ¹³C 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 ¹²C, ¹³C and ¹⁴C occur naturally on Earth with a percent of about 99%, 1% and 10⁻⁴%, ¹⁴C being a radioactive isotope with a half life of about 5730 years and having spin zero. Therefore, ¹⁴C 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 for correspond to the amplitude of probability to find the system on the states and, and one has and. Decimal notation on the energies corresponds to its binary elements for.

As before, three qubits dynamics is described by the equations

with and,

. The phases are defined as

, where decimal corresponds to its binary elements for.

For both cases, the energies are deduced from Equation (15).