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Physical Review A

Ensemble quantum-information processing by NMR: Implementation of gates and the creation of pseudopure states using dipolar coupled spins as qubits

T. S. Mahesh, Neeraj Sinha, K. V. Ramanathan, Anil Kumar

Published Jan 11, 2002• 23 citations
Publisher

Abstract

Quantum-information processing is carried out using dipolar coupled spins and high-resolution nuclear magnetic resonance (NMR). The systems chosen are the dipolar coupled methyl protons of ${\mathrm{CH}}_{3}\mathrm{CN}$ partially oriented in a liquid crystalline matrix yielding a two-qubit system and dipolar coupled ${}^{13}\mathrm{C}$ and methyl protons of ${}^{13}{\mathrm{CH}}_{3}\mathrm{CN}$ also partially oriented in the liquid crystalline matrix, yielding a three-qubit system. The dipolar coupled protons of oriented ${\mathrm{CH}}_{3}$ group are chemically and magnetically identical and their eigenstates can be divided into a set of quartet states (symmetric A) and a pair of doublet (E) states. We describe here a method for selectively retaining the magnetization of the symmetric states, yielding two and three qubit systems. We create pseudopure states using single-quantum-transition selective pulses and implement two- and three-qubit gates using one- and two-dimensional NMR.

DOI
10.1103/physreva.65.022312

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Theoretical Quantum Information Processing (QIP) has matured from the use of qubits to the use of qudits (systems having states >2). Whereas most of the experimental implementations have been performed using qubits, little experimental work has been carried out using qudits as yet. In this paper we demonstrate experimental realization of a qutrit system by nuclear magnetic resonance (NMR), utilizing deuterium (spin-1) nuclei partially oriented in liquid crystalline phase. Preparation of pseudopure states and implementation of unitary operations are demonstrated in this single-qutrit system, using transition selective pulses.
The enormous theoretical potential of Quantum Information Processing (QIP) is driving the pursuit for its practical realization by various physical techniques. Currently Nuclear Magnetic Resonance (NMR) has been the forerunner by demonstrating a majority of quantum algorithms. In NMR, spin systems consisting of coupled nuclear spins are utilized as qubits. In order to carry out QIP, a spin system has to meet two major requirements: (i) qubit addressability and (ii) mutual coupling among the qubits. It has been demonstrated that the magnitude of the mutual coupling among qubits can be increased by orienting the spin-systems in a liquid crystal matrix and utilizing the residual dipolar couplings. While utilizing residual dipolar couplings may be useful to increase the number of qubits, nuclei of same species (homonuclei) might become strongly coupled. In strongly coupled spin-systems, spins loose their individual identity of being qubits. We propose that even such strongly coupled spin-systems can be used for QIP and the qubit-manipulation can be achieved by transition-selective pulses. We demonstrate experimental preparation of pseudopure states, creation of maximally entangled states, implementation logic gates and implementation of Deutsch-Jozsa (DJ) algorithm in strongly coupled 2,3 and 4 spin systems. The energy levels of the strongly coupled 3 and 4 spin systems were obtained by using a Z-COSY experiment.
Nuclear magnetic resonance spectroscopy has demonstrated significant experimental progress toward the development of quantum computations. The developments so far have taken place mainly through the use of spin 12 nuclei. In this paper we describe the use of a spin 32 nucleus, oriented in a liquid crystal matrix for the creation of pseudopure states and the implementation of a complete set of two-qubit reversible logic gates using single-quantum transition-selective pulses, extending the range of practice of NMR toward quantum computation.
Physical implementation of quantum-information processing by liquid-state nuclear magnetic resonance, using weakly coupled spin-12\frac{1}{2}21​ nuclei of a molecule, is well established. Nuclei with spin>1/2\mathrm{spin}>1/2spin>1/2 oriented in liquid-crystalline matrices is another possibility. Such systems have multiple qubits per nuclei and large quadrupolar couplings resulting in well separated lines in the spectrum. So far, creation of pseudopure states and logic gates has been demonstrated in such systems using transition selective radio-frequency pulses. In this paper we report two developments. First, we implement a quantum algorithm that needs coherent superposition of states. Second, we use evolution under quadrupolar coupling to implement multiqubit gates. We implement the Deutsch-Jozsa algorithm on a spin-32\frac{3}{2}23​ (2 qubit) system. The controlled-NOT operation needed to implement this algorithm has been implemented here by evolution under the quadrupolar Hamiltonian. To the best of our knowledge, this method has been implemented for the first time in quadrupolar systems. Since the quadrupolar coupling is several orders of magnitude greater than the coupling in weakly coupled spin-12\frac{1}{2}21​ nuclei, the gate time decreases, increasing the clock speed of the quantum computer.
Residual dipolar couplings can be used to increase the number of qubits for quantum information processing. We have used molecules containing 3, 5 and 8 spins oriented in a liquid crystal matrix, and exploited the residual dipolar coupling to demonstrate quantum information processing in them. Transition assignment is performed using HET-Z-COSY experiment and qubit addressability is achieved by transition selective pulses. It is expected that using this protocol higher qubits can be achieved. For the implementations reported in this work, evolution under the internal Hamiltonian was not explored. It is however interesting to investigate how effectively the evolution under internal Hamiltonian can be manipulated to implement quantum algorithms in these systems. Recently an approach has been reported in this direction, where a new method of preparing pseudopure states in oriented systems by exciting selected multiple quantum using evolution under effective dipolar Hamiltonian, has been demonstrated [24].
A method for creating pseudopure states in NMR is described that combines spatial averaging and logical labeling techniques. This method uses one labeling qubit and any desired pseudopure state of any number of qubits can be created in the subsystem manifolds of the labeling qubit. The method uses a nonselective $\ensuremath{\pi}/2$ radio frequency pulse on all the nonlabeling qubits to equalize the populations of all the states within each subsystem, followed by a single transition-selective $\ensuremath{\pi}$ pulse on one of the transitions of the labeling qubit. The transverse magnetization created during the process is destroyed by a magnetic field gradient pulse, yielding a subsystem pseudopure state. The method is demonstrated experimentally on a system of three qubits.