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Journal of Experimental and Theoretical Physics Letters

Adder on ternary base elements for a quantum computer

V. E. Zobov, D. I. Pekhterev

Published May 1, 2009• 13 citations
Publisher

Abstract

A circuit consisting of elementary quantum logic operators has been proposed for an adder in the ternary number system. A sequence of RF magnetic field pulses has been found for its implementation by the nuclear magnetic resonance method on a chain of quadrupole nuclei with spin I = 1. The numerical simulation of the adder operation has been performed.

DOI
10.1134/s0021364009050105

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A circuit consisting of elementary quantum logic operators has been proposed for an adder in the ternary number system. A sequence of RF magnetic field pulses has been found for its implementation by the nuclear magnetic resonance method on a chain of quadrupole nuclei with spin I = 1. The numerical simulation of the adder operation has been performed.
It is shown theoretically that by the use of two radio-frequency fields of the same resonance frequency but with the different phases and directions the degeneracy of the energy spectrum of a spin system with I = 3/2 is removed. This leads to four non-degenerate spin states which can be used as a platform for quantum computing. The feasibility of quantum computing based on a pure (without DC magnetic fields) nuclear quadrupole resonance technique is investigated in detail. Various quantum logic gates can be constructed by using different excitation techniques allowing different manipulations with the spin system states. Three realizations of quantum logic gates are considered: the application of an additional magnetic field with the resonance frequency, the amplitude modulation of one of the applied RF fields by the resonance frequency field, and the level-crossing method. It is shown that the probabilities of the resonance transitions depend on the method of excitation and on the direction of the excitation field. Feasibility of quantum computing is demonstrated with the examples of constructing a controlled-NOT logic gate using the resonance excitation technique and SWAP and NOT2 logic gates using the level-crossing method.
In quantum information processing, realization of 3-qubit quantum operation is challenging especially when multiple logic operations are required to build non-trivial quantum circuits. Quantum half-adder is a 3-qubit circuit and in this work, a simple realization scheme for quantum half-adder operation is proposed. The 3-qubit quantum circuit is composed of a Toffoli operation and a controlled-NOT operation in succession. These two operations are shown to be achievable only by two transition pulses in nuclear magnetic resonance based quantum computing at room temperature. In our proposed scheme, three weakly coupled spin 1/2 nuclei of a molecule serve as three qubits. The time-independent internal Hamiltonian of the system allows us to obtain the desired multi-qubit operation. The simple structure of the spin 1/2 systems of our scheme make it easy to be implemented in experiments. Simulation on fidelity estimation reveals that more than 96% fidelity is achievable in worst case scenario.
A quantum circuit has been proposed for the algorithm for finding the permutation order on two qudits with the number of levels d 1 and d 2. The sequence of the RF pulses for implementing the algorithm on two quadrupole nuclei I 1 = 7/2 (d 1 = 8) and I 2 = 3/2 (d 2 = 4) has been calculated and the algorithm has been numerically simulated. A method for preparing pseudopure states has been presented. A comparison with the implementation of the algorithm by NMR methods on five qubits has been performed.
The advantages of using quantum systems for performing many computational tasks have already been established. Several quantum algorithms have been developed which exploit the inherent property of quantum systems such as superposition of states and entanglement for efficiently performing certain tasks. The experimental implementation has been achieved on many quantum systems, of which nuclear magnetic resonance has shown the largest progress in terms of number of qubits. This paper describes the use of a spin-7/2 as a three-qubit system and experimentally implements the half-adder and subtractor operations. The required qubits are realized by partially orienting 133Cs{}^{133}\mathrm{Cs}133Cs nuclei in a liquid-crystalline medium, yielding a quadrupolar split well-resolved septet. Another feature of this paper is the proposal that labeling of quantum states of system can be suitably chosen to increase the efficiency of a computational task.
Ternary quantum logic plays a very important role for building high speed and efficient futuristic computers. It has several advantages over classical computing and binary quantum circuits. In this paper, the realization of basic ternary circuits for adder/subtractor, encoder and priority encoder are proposed and designed. These circuits are very essential for the construction of various computational units of quantum computers and other complex computational systems. Design of reversible circuits can be improved by reducing the quantum circuit cost. This paper uses some elementary components and typical ternary gates (generalized ternary gate, M-S gate etc.) to perform arithmetic addition, subtraction and encoding operations. Finally, it evaluates the optimum cost for each circuit.