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Revista Mexicana de Física

Effects on the quantum not and controlled-not gates of a modular magnetic field in the z-direction in a chain of nuclear spin system

Gustavo V. López, Manuél Ávila

Published Aug 15, 2011• 0 citations
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Abstract

We study the simulation of a single qubit rotation and Controlled-Not gate in a solid state one-dimensional chain of nuclear spins system interacting weakly through an Ising type of interaction with a modular component of the magnetic field in the z-direction, characterized by Bz (z,t) = Bo(z) cos dt. These qubits are subjected to electromagnetic pulses which determine the transition in the one or two qubits system. We use the fidelity parameter to determine the performance of the Not (N) gate and Controlled-Not (CNOT) gate as a function of the frequency parameter S. We found that for |d| 10-3 MHz, these gates still have good fidelity.

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We study the simulation of a single qubit rotation and Controlled-Not gate in a solid state one-dimensional chain of nuclear spins system interacting weakly through an Ising type of interaction with a modular component of the magnetic field in the z-direction, characterized by Bz (z,t) = Bo(z) cos dt. These qubits are subjected to electromagnetic pulses which determine the transition in the one or two qubits system. We use the fidelity parameter to determine the performance of the Not (N) gate and Controlled-Not (CNOT) gate as a function of the frequency parameter S. We found that for |d| 10-3 MHz, these gates still have good fidelity.
We study the simulation of a single qubit rotation and Controlled-Not gate in a solid state one-dimensional chain of nuclear spins system interacting weakly through an Ising type of interaction with a modular component of the magnetic field in the z-direction, characterized by Bz(z,t)=Bo(z)cos⁡δtB_z(z,t)=Bo(z)\cosδtBz​(z,t)=Bo(z)cosδt. These qubits are subjected to electromagnetic pulses which determine the transition in the one or two qubits system. We use the fidelity parameter to determine the performance of the Not (N) gate and Controlled-Not (CNOT) gate as a function of the frequency parameter δδδ. We found that for ∣δ∣≤10−3MHz|δ|\le 10^{-3} MHz∣δ∣≤10−3MHz, these gates still have good fidelity.
We present the study of a quantum Controlled-Controlled-Not gate, implemented in a chain of three nuclear spins weakly Ising interacting between all of them, that is, taking into account first and second neighbor spin interactions. This implementation is done using a single resonant pi\\pipi-pulse on the initial state of the system (digital and superposition). The fidelity parameter is used to determine the behavior of the CCN quantum gate as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant (J′/JJ'/JJ′/J). We found that for J′/Jge0.02J'/J\\ge 0.02J′/Jge0.02 we can have a well defined CCN quantum gate.
We present the study of a quantum Controlled-Controlled-Not gate, implemented in a chain of three nuclear spins weakly Ising interacting between all of them, that is, taking into account first and second neighbor spin interactions. This implementation is done using a single resonant πππ-pulse on the initial state of the system (digital and superposition). The fidelity parameter is used to determine the behavior of the CCN quantum gate as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant (J′/JJ'/JJ′/J). We found that for J′/J≥0.02J'/J\ge 0.02J′/J≥0.02 we can have a well defined CCN quantum gate.
We study the phenomenon of decoherence during the operation of one qubit transformation, controlled-not (CNOT) and controlled-controlled-not (C2NOT) quantum gates in a quantum computer model formed by a linear chain of three nuclear spins system. We make this study with different type of environments, and we determine the associated decoherence time as a function of the dissipative parameter. We found that the dissipation parameter to get a well defined quantum gates (without significant decoherence) must be within the range of . We also study the behavior of the purity parameter for these gates and different environments and found linear or quadratic decays of this parameter depending on the type of environments.
We study numerically the non-resonant effects on four-spin molecules at room temperature with the implemented quantum controlled-not gate and using the 2π k method. The four nuclear spins in each molecule represent a four-qubit register. The qubits interact with each other through Ising-type interaction which is characterized by the coupling constant J a , b . We study the errors on the reduced density matrix as a function of the Rabi frequency, Ω, using the 2π k method and when all the coupling constants are equal or when one of them is different from the others.
The effect of frequency noise on correct operation of the multiple-control Toffoli, Fredkin, and Peres gates has been discussed. In the framework of the Ising model, the energy spectrum of a chain of atoms with nuclear spins one-half in a spinless semiconductor matrix has been obtained, and allowed transitions corresponding to the operation algorithm of these gates have been determined. The fidelities of the obtained transitions were studied depending on the number of control qubits and parameters of the radio-frequency control pulses. It has been shown that correct operation of the Toffoli and Fredkin gates does not depend on the number of control qubits, while the Peres gate fidelity decreases significantly with the increasing number of control signals. The calculated ratios of the Larmor frequency to the exchange interaction constant correspond with the results of other studies.