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Journal of Experimental and Theoretical Physics

Selective control of the states of multilevel quantum systems using nonselective rotation operators

V. E. Zobov, V. P. Shauro

Published Jan 1, 2009• 3 citations
Publisher

Abstract

We have calculated the sequences of nonselective rotation operators separated by intervals of free evolution that perform selective rotations between adjacent levels in systems with three, four, five, and six nonequidistant levels. We have numerically simulated the realization of the calculated sequences for quadrupole nuclei with corresponding spins controlled by intense nonselective radio-frequency (RF) pulses and investigated the dependences of the realization error on the parameters of external and internal interactions. To reduce the error when the RF field is not strong enough, we have found composite nonselective RF pulses consisting of five simple ones. We show that the error of the composite selective rotation operator can be reduced signifi- cantly in comparison to the error of a simple single selective pulse.

DOI
10.1134/s1063776109010026

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We have calculated the sequences of nonselective rotation operators separated by intervals of free evolution that perform selective rotations between adjacent levels in systems with three, four, five, and six nonequidistant levels. We have numerically simulated the realization of the calculated sequences for quadrupole nuclei with corresponding spins controlled by intense nonselective radio-frequency (RF) pulses and investigated the dependences of the realization error on the parameters of external and internal interactions. To reduce the error when the RF field is not strong enough, we have found composite nonselective RF pulses consisting of five simple ones. We show that the error of the composite selective rotation operator can be reduced signifi- cantly in comparison to the error of a simple single selective pulse.
Elementary logical operators (selective rotation, Fourier transform, controllable phase shift, and SUM gate) are considered for a quantum computer based on three-level systems (qutrits) represented by nuclear spins I = 1 under nuclear magnetic resonance conditions. The computer simulation of the realization of these operators by means of simple and composite selective radiofrequency (RF) pulses and optimized RF pulses is performed. The time dependence of the amplitude of last pulses is found by numerical optimization at different durations. Two variants are proposed for realization of a two-qutrit SUM gate by using one-qutrit or two-qutrit optimized RF pulses. The calculated time dependences of realization errors were used to study the time optimality of different methods for obtaining gates, proposed earlier and in this paper. The advantages and disadvantages of each of the methods are evaluated for different values of physical parameters.
To experimentally realize quantum computations on d -level basic elements (qudits) at d > 2, it is necessary to develop schemes for the technical realization of elementary logical operators. We have found sequences of selective rotation operators that represent the operators of the quantum Fourier transform (Walsh-Hadamard matrices) for d = 3–10. For the prime numbers 3, 5, and 7, the well-known method of linear algebra is applied, whereas, for the factorable numbers 6, 9, and 10, the representation of virtual spins is used (which we previously applied for d = 4, 8). Selective rotations can be realized, for example, by means of pulses of an RF magnetic field for systems of quadrupole nuclei or laser pulses for atoms and ions in traps.