500568\db996088-4b89-4f7d-b799-89e0a73465ce.jpg width=98.6100028991699 height=31.825 /> represents the ascend operator () or the descend operator (). The Hamiltonian is diagonal in the basis with (zero for the ground state and one for the exited state). The action of the spin operators on its respective qubit is given by, , and. The eigenvalues of in this basis are given by
The elements of this basis forms a register of N-qubits with a total number of registers which is the dimensionality of our Hilbert space. The allowed transition of one state to another one is gotten by choosing the angular frequency of the RF-field, , as the associated angular frequency due to the energy difference of these two levels, and by choosing the normalized evolution time with the proper time duration (so called RF-field pulse). The set of selected pulses defines the quantum gate we want to study with this quantum computer.
3. System-Environment Interaction Models
Now, to deal with the non ideal situation where the effect of the environment is taken into account, we make use of the Lindblad type master equation for the evolution of the reduced density matrix
where the first part on the right side denotes the usual von Neuman unitary evolution of the reduced density matrix, and the second term represents the Lindblad part (non unitary) evolution. This second term has different expression for different consideration of the system-environment interaction. For the qubits interacting independently with the environment (case (A)), this term has the following form 
where is the dissipative parameter associated to the jth-qubit.
For the pure dephasing interaction case, where the qubits independently dephase to their respective bath with a dephasing rate, the Lindblad term is given by
For the independent-qubit-correlated case (qubits interact with the environment collectively), the Lindblad operator is written as
where one has that is the decay rate of case (A). In this case, the decay of the state of a qubit has an effect on the other qubits.
For the qubit-correlated and dephasing case, with as the decay rate of the correlated dephasing, the Lindblad operator is given by
In this case, the decay of one qubit affects too the other qubits.
The dynamical system for each case for the reduced density matrix elements is deduced from Equation (7) as
where and represent the elements of the basis of the Hilbert space,andand naming the ground state as the first state of the system. In our case, one has that, the dimensionality of our Hilbert space is eight, and the explicit equations for the dynamical system of each case are given in the appendix.
4. Simulation Results
Our registers are made up of three qubits with, or written them with decimal notation, , and so on (do not confuse with the type of environment). The parameters used for our simulation are taken from  and are given (in units of) as
Using the assumption that the environment acts homogeneously on the qubits, the damping parameter can be the same for each qubit, and the damping parameter for correlated cases at second neighbors can be one order of magnitude weaker that at first neighbors. Thus, the dissipative coefficients appearing for the cases (A), (B), (C), and (D) are taken of the following way
where is the free common parameter which takes into account the interaction with the environment. The reduced density matrix is then made up of 8 × 8 complex elements, and if the initial state is always taken as the ground state of the quantum state, this means that the initial reduced density matrix has the values and for.
4.1. Single Qubit Rotation
We have selected the transition (or in decimal notation:) as an example of single qubit rotation (corresponding to the NOT quantum gate), and the NOT-reduced density matrix after a -pulse with resonant frequency would be such that and for. In addition, we allow the system to run for almost two an a half -pulses more to see how the NOT gate evolves. Figure 1 shows the decoherence behavior (damping of the non diagonal matrix element) for the cases (A), (B), (C) and (D), using different values for the damping parameter. In it, we can see a revival in the coherence with smaller amplitude, which is due to the three and a half pulses applied to the system. The effect of the environment is to reduce the amplitude of the coherent terms involved in the transitions, and to create new superposition of other states. Once enough time has passed, the destruction of the expected superposition will be absolute and new states take the entire probability of the system, like the state related to the element in Figure 2.
The Figure 2 shows the behavior of the diagonal reduced density matrix elements when dissipation is switched on, with. Since one must have that, the damping of the matrix elements and (destruction of the transition of the states) brings about the excitation of other matrix elements as shown on left hand side figures. Our simulation also show that the environment generating dephasing, described by cases B and D are less harmful to the system than the environments related to cases (A) and (C). The system interacting with environments B and C has a faster recovery when the dissipation parameter goes weaker.
Figure 1. Non diagonal element for (1), (2), (3), and (4).
Figure 2. Diagonal elements of the reduced density matrix for.
To characterize the decoherence behavior of the off diagonal element of the reduced density matrix, assume that one defines the decay characteristic parameter as as the time when the matrix element has a value of on the first pulse. Then, Figure 3 shows the dependence of this characteristic parameter as a function of the dissipative parameter for our four cases. The behavior of this parameter as a function of the dissipative parameter can be fixed by a linear relation, , where for the independent cases (A) and (C), and for the dephasing cases (B) and (D).
Other parameter which can help us to see the departure from the ideal () behavior could be the distance between the non diagonal matrix elements of the ideal () and the dissipative cases (, for). Figure 4 shows the behavior of this distance for different values of indexed by (1), (2), (3) and (4). Of course, this distance increases as increases because the decoherence is taking place deeper. The maximum separation corresponds to the maximum on the coherence behavior as shown in Figure 1.
4.2. Controlled-Not (CNOT) Quantum Gate
To get the CNOT quantum gate starting from the ground state, one applies a -pulse between this state and the state, with resonant frequency, to get the superposition state. Then, one applies a resonant -pulse between the states and, with resonant frequencyto get the final desired state which means that the expected CNOT density matrix would be such that, and all the other elements are equal to zero. In addition, one allows the system to have two and a half more resonant -pulses to have a better look of the CNOT behavior. Figure 5 shows the decoherence behavior (damping of the non diagonal matrix elements) for the cases (A), (B), (C) and (D), using different the damping parameters. One observes from these figures that there is not significant difference between independent and correlated-independent, and between dephasing and correlated-dephasing cases (similar behavior is observed for the other non diagonal matrix elements).
We note the peculiar behavior of the term for the cases (A) and (C) of Figure 5, for a relatively high dissipation (labeled (1)). Before this superposition occurs, other elements of the density matrix have already been exited due to the interaction with the environment and the condition of. So, the other excited states seems to have an important influence to slow the formation of the superposition, characterized by the term. As the dissipation parameter goes weaker, a regular behavior is observed. We can also see on Figure 5 that for short times, other states get exited besides the ones involved in the CNOT gate, therefore there are other states also overlapped with the states and. A revival in the superposition of and can be seen when enough of time has passed so that the other states excited by the environment have also went down (curve labeled with (1)). Figure 6 shows the diagonal reduced density matrix elements with, for the four cases into consideration. Cases (A) and (C) shows the increasing of other diagonal matrix elements to keep, corresponding to excitation of the state.
Figure 3. Characteristic time as a function of dissipation parameter.
Figure 4. Distance between the ideal and dissipative coherent term for different values of as a function of time.
Figure 5. Behavior of non diagonal element for (1), (2), (3), and (4).
As we observed with the NOT quantum gate before, in our simulations we observed for the CNOT gate that the dependence of the characteristic decay parameter with respect the dissipation parameter is linear for the non diagonal elements, and, although their parameters a and b have different values for different non diagonal elements. However, these values are almost the same for the cases (A)-(C) and (B)-(D). In addition, the behavior of the distance for cases for the non diagonal matrix elements is again very similar to that one found for the NOT quantum gate (Figure 5), that is, we observed and increasing of this distance as the dissipation gets stronger.
4.3. Controlled-Controlled-Not (C2NOT) Quantum Gate
To get the C2NOT quantum gate starting from the ground state, one applies a -pulse between this state and the state with resonant frequency to get the superposition state. Then, one applies a resonant -pulse between the states and the state with resonant frequency to get the desired superpositional state. Finally, C2NOT operation is achieved through a pulse between the state and the state, getting the state. In addition, we allow the system to evolve for another almost two an a half pulses to see the behavior of the C2NOT. The expected C2NOT density matrix would be such that and all the other elements equal to zero. Figure 7 shows the behavior of the non diagonal matrix elements for the cases (A), (B), (C) and (D), using different damping parameters. As in the CNOT gate, a similar strange behavior is seen for the coherent terms which appears later in the gate (and) for the cases (A) and (C) at relatively high
Figure 6. Diagonal elements of the reduced density matrix for the CNOT gate and.
Figure 7. Non diagonal reduced density matrix element for (1), (2), (3), and (4).
dissipation parameter, that is, the state is excited for strong dissipation.
Figure 8 shows the diagonal reduced density matrix elements with, for the four cases into consideration. Cases (A) and (C) showing the excitation of other diagonal elements to keep.
As for the NOT and CNOT quantum gates, in our simulation we observe a similar expected linear dependence of the characteristic decay parameter with respect the dissipative parameter for the non diagonal elements, although the parameters a and b have different values for different non diagonal elements. The behavior of the difference, between the non diagonal matrix elements of their ideal values and their dissipative values and for the cases
is again similar as the other gates.
5. Purity Calculations
The purity function, , is a measure of how close a quantum system is from its description as a pure state quantum system and varies between 1 and (d the dimensionality of the density matrix). This function may decay with the decoherence since the system may move away from an initial pure state. Therefore, this function can be used to characterize the environment. Figure 9 shows the behavior of the purity for the CNOT gate using several dissipative parameters (the behavior of the purity is similar for the NOT and C2NOT quantum gates). As one ca see from these figures, Independent environment is characterized by an initial linear damping (exponential) in the purity function, meanwhile dephasing damping is characterized by an initial quadratic damping (Gaussian) in the purity function. This an be understood because the dephasing case has weaker effect on the decoherence of the system, as can be seen from
Figure 8. Diagonal elements of the reduced density matrix for the C2NOT gate and.
Figure 9. Purity for the CNOT quantum gate.
Figure 5, where the maximum values reached by the non diagonal matrix density elements are bigger for dephasing case than for independent case. Purity increases its value after the first decay for high damping values () and for the independent case since for this value a particular state is exited (trying to keep) as seen in Figure 6.
Within the Markov approximation for the study of quantum discrete system with environment, we have solved numerically the master equation for the reduced density matrix associated to our linear chain of three nuclear spin quantum computer interacting with the environment, and we have made the simulation of a particular single qubit transition (NOT gate), CNOT and C2NOT quantum gates operating within this dissipative environment. Within the validity of this approximation, the coherence of these quantum logic gates have been determined, and we have calculated the associated decoherence time (using the first pulse applied to the system), , as a function of a common dissipative parameter (all the dissipative parameters appearing in the Lindblad part of the master equation were chosen proportional a single one), and it was found the expected linear dependence of this coherence time with respect this dissipative parameter, but with different decay rates for the non diagonal matrix elements and gates. The value of the dissipative parameter for the environment not to affect the performance of these quantum gates (within two cycles and a half after their formation) was found to be. We used the purity parameter to study its behavior of this chain of three nuclear spin system interacting with the environment, and we have found an initial linear (or exponential) damping of the purity parameter for the independent case, and an initial quadratic (or Gaussian) damping on its behavior for the dephasing case, giving us an indication of strong (linear) or weak (quadratic) effect of the environment in our system. For very strong dissipation, we found that purity may increase because, the condition on the density matrix, sometimes implies strong excitation of other state involved in the dynamics, causing the system to try to return to a pure quantum state description.
The evolution equation for the density matrix elements are given from Equation (12) by
Making the following definition
Von Neuman (vN) Part
Case A: Independent
Case B: Pure Dephasing
Case C: Independent Correlated
The Lindblad expression is given by
where is as the case A, and it can be written as
where the the independent part A corresponds to the case A above and the correlated part is
Case D: Dephasing Correlated
The Lindblad term is
with being the as the case B (Pure dephasing), and one has. This expression can be written as
where is the Lindblad term of the case B (pure dephasing), and the other term is given by