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Physica B Condensed Matter

Dynamics of a Control-Not gate for a quantum system of two weakly interacting spins

G. P. Berman, David K Campbell, G. D. Doolen, Gustavo V. López, V. I. Tsifrinovich

Published Sep 1, 1997• 10 citations
Publisher

Abstract

Abstract We present the results of computer simulations of dynamics of the quantum Control-Not gate for the system of two spins connected by a weak Ising interaction. It is demonstrated that, for both “digital” and “superposition” initial conditions, 3 resonant π-pulse can provide an adequate implementation of the quantum Control-Not gate. We also consider the influence of slightly non-resonant π-pulse and noise on the dynamics of the quantum Control-Not gate.

DOI
10.1016/s0921-4526(97)00434-1

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Abstract We present the results of computer simulations of dynamics of the quantum Control-Not gate for the system of two spins connected by a weak Ising interaction. It is demonstrated that, for both “digital” and “superposition” initial conditions, 3 resonant π-pulse can provide an adequate implementation of the quantum Control-Not gate. We also consider the influence of slightly non-resonant π-pulse and noise on the dynamics of the quantum Control-Not 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 investigate numerically a single-electromagnetic-pulse implementation of a quantum control-not (CN) gate for an ensemble of Ising spin systems at room temperature. For an ensemble of four-spin ``molecules'' we simulate the time evolution of the density matrix for both digital and superpositional initial conditions. Our numerical calculations confirm the feasibility of implementation of a quantum CN gate in this system at finite temperature, using a single electromagnetic \ensuremath{\pi} pulse. We also study the quantum dynamics of creating entangled states in a macroscopic paramagnetic spin system at low temperatures, and compare the related quantum and corresponding classical dynamics. In the quantum case, one can create entangled states using a resonant interaction between the spin system and the electromagnetic \ensuremath{\pi} pulse. In the classical limit, the interaction of the spin system with the same pulse becomes nonresonant and the corresponding classical dynamics differs significantly from the quantum dynamics. This difference in the behavior of the macroscopic magnetization can be used in experiments as an indication of the existence of entangled states.
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.
The physical implementation of the quantum Control-Not gate for a two-spin system is investigated numerically. The concept of a generalized quantum Control-Not gate, with arbitrary phase shift, is introduced. It is shown that a resonant πππ-pulse provides a simple example of a generalized quantum Control-Not gate.
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.