Numerical simulation of a controlled–controlled-not (CCN) quantum gate in a chain of three interacting nuclear spins system

Abstract

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 π-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 ′ /J). We found that for J ′ /J ≥ 0.02 we can have a well defined CCN quantum gate.

Study of Decoherence of Elementary Gates Implemented in a Chain of Few Nuclear Spins Quantum Computer Model

“…We note the peculiar behavior of the term 34  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 Tr = 1  .…”

“…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 34 π  ) 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).…”

“…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 Tr = 1  . So, the other excited states seems to have an important influence to slow the formation of the superposition, characterized by the term 34  .…”

“…One of the used approaches for quantum discrete system is described in [30], where it was used for studying sudden death of entanglement of two qubits. We will use this approach for our study of decoherence of several quantum gates during operation in a quantum computer model made up of a linear chain of three paramagnetic atoms with nuclear spin one half [31,34,36]. In this work, we are interested in determine the decoherence of a single qubit rotation, quantum controlled-not (CNOT), and quantum controlled-controllednot (C 2 NOT) gates during their implementation on this quantum computer model.…”

“…We describe the model and the Hamiltonian of our quantum system, and we must point out that, although this Hamiltonian will be time explicitly dependent, if we consider weak interaction between our system and the environment (the characteristic times of the quantum system are much longer than those of the environment) as a first approximation, the above mentioned Markovian-Lindblad master type equation can be still used for our study [3,32,33]. On the other hand, even this model for solid state quantum computer has not been built yet, it has been very useful for theoretical studies about implementation of quantum gates and quantum algorithms [34][35][36] which can be extrapolated to other solid state quantum computers [37-39].…”