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The Journal of Chemical Physics

Nuclear magnetic resonance quantum logic gates using quadrupolar nuclei

Anatoly K. Khitrin, B. M. Fung

Published Apr 22, 2000• 81 citations
Publisher

Abstract

The use of nuclei with spin >1/2 and quadrupolar splitting of the spin energy levels has certain advantages over systems of interacting spins 1/2 for building quantum logic gates. Experimental demonstration has been made using 23Na (spin 3/2) NMR in a lyotropic liquid crystal system. The pulse sequences for establishing effective pure quantum states are given, and the resulting NMR spectra are presented.

DOI
10.1063/1.481293

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The use of nuclei with spin >1/2 and quadrupolar splitting of the spin energy levels has certain advantages over systems of interacting spins 1/2 for building quantum logic gates. Experimental demonstration has been made using 23Na (spin 3/2) NMR in a lyotropic liquid crystal system. The pulse sequences for establishing effective pure quantum states are given, and the resulting NMR spectra are presented.
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.
In this thesis, we apply quantum logic gates to a two-qubit register using the techniques of nuclear magnetic resonance (NMR spectroscopy). We create a lyotropic solution of sodium decyl sulfate (SDS), deuterium oxide (D2O), and decanol in the nematic phase, which functions as our quantum register. Using sequences of single-quantum selective pulses, we generate pseudopure initial states. Using SR, a MatLab-based spin response simulator, we check that the pulse sequences produce the appropriate pseudopure states. We then apply a CNOT quantum logic gate to each initial state. Each pseudopure state reacts as expected to the CNOT gate: while some extra excitations occur, they are generally small compared to the rest of the spectrum. These extra excitations may be the result of imperfect phase cycling in our pseudopure pulse sequences, which would leave non-zero coherences between states. These coherences could interfere with the resulting NMR spectra, producing the addition observed excitations.
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.
Nuclear magnetic resonance is viewed as an important technique for the implementation of many quantum information algorithms and protocols. Although the most straightforward approach is to use the two-level system composed of spin 1/2 nuclei as qubits, quadrupolar nuclei, which possess a spin greater than 1/2, are being used as an alternative. In this study, we show some unique features of quadrupolar systems for quantum information processing, with an emphasis on the ability to execute efficient quantum state tomography (QST) using only global rotations of the spin system, whose performance is shown in detail. By preparing suitable states and implementing logical operations by numerically optimized pulses together with the QST method, we follow the stepwise execution of Grover's algorithm. We also review some work in the literature concerning the relaxation of pseudo-pure states in spin 3/2 systems as well as its modelling in both the Redfield and Kraus formalisms. These data are used to discuss differences in the behaviour of the quantum correlations observed for two-qubit systems implemented by spin 1/2 and quadrupolar spin 3/2 systems, also presented in the literature. The possibilities and advantages of using nuclear quadrupole resonance experiments for quantum information processing are also discussed.
Nuclear magnetic resonance is viewed as an important technique for the implementation of many quantum information algorithms and protocols. Although the most straightforward approach is to use the two-level system composed of spin 1 nuclei as qubits, quadrupolar nuclei, which possess a spin greater than 1 , are being used as an alternative. In this study, we show some unique features of quadrupolar systems for quantum information processing, with an emphasis on the ability to execute efficient quantum state tomography (QST) using only global rotations of the spin system, whose performance is shown in detail. By preparing suitable states and implementing logical operations by numerically optimized pulses together with the QST method, we follow the stepwise execution of Grover’s algorithm. We also review some work in the literature concerning the relaxation of pseudo-pure states in spin 3 systems as well as its modelling in both the Redfield and Kraus formalisms. These data are used to discuss differences in the behaviour of the quantum correlations observed for two-qubit systems implemented by spin 1 and quadrupolar spin 3 systems, also presented in the literature. The possibilities and advantages of using nuclear quadrupole resonance experiments for quantum information processing are also discussed.
H– 13 1 C heteronuclear dipolar couplings are used to produce the NMR (nuclear magnetic resonance) version of a two bit controlled-NOT quantum logic gate. This gate is coupled with the Hadamard gate to complete a circuit which generates the Einstein–Podolsky–Rosen (EPR) state which is the maximally entangled state of a pair of spins. The EPR state is crucial for the potential exponential speed advantage of quantum computers over their classical counterparts. We sample the deviation density matrix of the two spin system to verify the presence of the EPR state. EPR state lifetimes are also measured with this technique, thereby demonstrating the viability of liquid crystals as a platform for quantum computing.