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arXiv:quant-ph/0610224 · Physical Review A

Quantum-information processing using strongly dipolar coupled nuclear spins

T. S. Mahesh, Dieter Suter

Published Dec 18, 2006• 42 citations
arXivHTMLPDF

Abstract

Dipolar coupled homonuclear spins present challenging, yet useful systems for quantum-information processing. In such systems, the eigenbasis of the system Hamiltonian is the appropriate computational basis and coherent control can be achieved by specially designed strongly modulating pulses. In this paper we describe the first experimental implementation of the quantum algorithm for numerical gradient estimation by nuclear magnetic resonance, using the eigenbasis of a four spin system.

DOI
10.1103/physreva.74.062312

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Physical Review A•Jan 11, 2002•23 citations
Dipolar coupled homonuclear spins present challenging, yet useful systems for quantum-information processing. In such systems, the eigenbasis of the system Hamiltonian is the appropriate computational basis and coherent control can be achieved by specially designed strongly modulating pulses. In this paper we describe the first experimental implementation of the quantum algorithm for numerical gradient estimation by nuclear magnetic resonance, using the eigenbasis of a four spin system.
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.
Fifty years of developments in nuclear magnetic resonance (NMR) have resulted in an unrivaled degree of control of the dynamics of coupled two-level quantum systems. This coherent control of nuclear spin dynamics has recently been taken to a new level, motivated by the interest in quantum information processing. NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. This article surveys and summarizes a broad variety of pulse control and tomographic techniques which have been developed for, and used in, NMR quantum computation. Many of these will be useful in other quantum systems now being considered for the implementation of quantum information processing tasks.
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 three-qubit 13C^{13}\mathrm{C}13C solid-state nuclear magnetic resonance (NMR) system for quantum-information processing, based on the malonic acid molecule, is used to demonstrate high-fidelity universal quantum control via strongly modulating radio-frequency pulses. This control is achieved in the strong-coupling regime, in which the time scales of selective qubit addressing and of two-qubit interactions are comparable. State evolutions under the internal Hamiltonian in this regime are significantly more complex, in general, than those of typical liquid-state NMR systems. Moreover, the transformations generated by the strongly modulating pulses are shown to be robust against the types of ensemble inhomogeneity that dominate in the employed molecular crystal system. The secondary focus of the paper is upon detailed characterization of the malonic acid system. The internal Hamiltonian of the qubits is determined through spectral simulation. A pseudopure state preparation protocol is extended to make a precise measurement of the dephasing rate of a three-quantum coherence state under residual dipolar interactions. The spectrum of intermolecular 13C−−13C^{13}\mathrm{C}--^{13}\mathrm{C}13C−−13C dipolar fields in the crystal is simulated, and the results compared with single-quantum dephasing data obtained using appropriate refocusing sequences. We conclude that solid-state NMR systems tailored for quantum-information processing have excellent potential for extending the investigations begun in the liquid-state systems to a greater number of qubits.
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 CH3CN{\mathrm{CH}}_{3}\mathrm{CN}CH3​CN partially oriented in a liquid crystalline matrix yielding a two-qubit system and dipolar coupled 13C{}^{13}\mathrm{C}13C and methyl protons of 13CH3CN{}^{13}{\mathrm{CH}}_{3}\mathrm{CN}13CH3​CN also partially oriented in the liquid crystalline matrix, yielding a three-qubit system. The dipolar coupled protons of oriented CH3{\mathrm{CH}}_{3}CH3​ 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.