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Physical Review A

Quantum-state tomography for quadrupole nuclei and its application on a two-qubit system

Fábio Aurélio Bonk, Roberto S. Sarthour, Eduardo Ribeiro DeAzevedo, J.D. Bulnes, Gerson L. Mantovani, Jair C. C. Freitas, Tito José Bonagamba, Alberto Passos Guimarães, Ivan S. Oliveira

Published Apr 29, 2004• 42 citations
Publisher

Abstract

A method for performing quantum state tomography for quadrupole nuclei is presented in this paper. First, it is shown that upon appropriate phase cycling, the nuclear-magnetic-resonance (NMR) intensities of quadrupole nuclei depend only on diagonal elements of the density-matrix. Thus, a method for obtaining the density-matrix elements, which consists of dragging off-diagonal elements into the main diagonal using fine phase-controlled selective radio-frequency pulses, was derived. The use of the method is exemplified through $^{23}\mathrm{Na}$ NMR (nuclear spin $I=3∕2$) in a lyotropic liquid crystal at room temperature, in three applications: (a) the tomography of pseudopure states, (b) the tomography of the quadrupole free evolution of the density matrix, and (c) the unitary state evolution of each qubit in the system over the Bloch sphere upon the application of a Hadamard gate. Further applications in the context of pure NMR and in the context of quantum information processing, as well as generalizations for higher spins, are discussed.

DOI
10.1103/physreva.69.042322

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A method for performing quantum state tomography for quadrupole nuclei is presented in this paper. First, it is shown that upon appropriate phase cycling, the nuclear-magnetic-resonance (NMR) intensities of quadrupole nuclei depend only on diagonal elements of the density-matrix. Thus, a method for obtaining the density-matrix elements, which consists of dragging off-diagonal elements into the main diagonal using fine phase-controlled selective radio-frequency pulses, was derived. The use of the method is exemplified through 23Na^{23}\mathrm{Na}23Na NMR (nuclear spin I=3∕2I=3∕2I=3∕2) in a lyotropic liquid crystal at room temperature, in three applications: (a) the tomography of pseudopure states, (b) the tomography of the quadrupole free evolution of the density matrix, and (c) the unitary state evolution of each qubit in the system over the Bloch sphere upon the application of a Hadamard gate. Further applications in the context of pure NMR and in the context of quantum information processing, as well as generalizations for higher spins, are discussed.
We discuss the applications of Nuclear Magnetic Resonance (NMR) to quantum information processing, focusing on the use of quadrupole nuclei for quantum computing. Various examples of experimental implementation of logic gates are given and compared to calculated NMR spectra and their respective density matrices. The technique of Quantum State Tomography for quadrupole nuclei is briefly described, and examples of measured density matrices in a two-qubit I = 3/2 spin system are shown. Experimental results of density matrices representing pseudo-Bell states are given, and an analysis of the entropy of theses states is made. Considering an NMR experiment as a depolarization quantum channel we calculate the entanglement fidelity and discuss the criteria for entanglement in liquid state NMR quantum information. A brief discussion on the perspectives for NMR quantum computing is presented at the end.
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.
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.
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.
We report a detailed study of the longitudinal relaxation of several coherent states on the nuclear-magnetic-resonance (NMR) two-qubit quadrupole system, 23Na{}^{23}\mathrm{Na}23Na (spin 3/2) in a lyotropic liquid-crystal system at room temperature. Relaxation of pseudopure states, simulated pseudo-Bell states and Hadamard states were investigated. The coherence of superposition states was verified by applying the respective quantum ``reading'' operators and comparing the recovered state with the original one. The degree of recovery varies between $\ensuremath{\approx}70%$ and $\ensuremath{\approx}100%,$ depending on the time length of the gate, as compared to relaxation times. Spin-lattice relaxation results follow a recently proposed multiexponential model that includes mixed magnetic dipolar and electric quadrupolar interactions. Relaxation curves are governed by initial conditions, and from the experimental curves individual transition rates are derived. The transverse relaxation time constant T2{T}_{2}T2​ is found not to depend on the initial distribution of populations. This is a systematic study of relaxation in the context of NMR quantum computing.
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.