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Process Tomography of a nuclear quadrupolar spin quantum processor: Characterization of quantum algorithms and non-Markovian decoherence

Hermann Kampermann

Published Jun 16, 2008• 0 citations
OpenAlex

Abstract

In the present work the first virtual 2 Qubit solid state NMR quantum processor, using a spin- 3/2 Na quadrupolar nucleus in a NaNO3 single crystal, is presented. Methods from liquid NMR quantum information processing, like strongly modulating pulses, are adopted to the quadrupolar spin system to implement the unitary operations and to generate pseudo pure states. Experimentally a set of quantum algorithms as well as the decoherence process is characterized by quantum process tomography which is developed for the quadrupolar spin system. A detailed analyses of the basic conditions of the experimental quantum process tomography procedure is presented, showing that even in such non-Markovian systems this method is valid.

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Physical Review A•Apr 29, 2004•42 citations
In the present work the first virtual 2 Qubit solid state NMR quantum processor, using a spin- 3/2 Na quadrupolar nucleus in a NaNO3 single crystal, is presented. Methods from liquid NMR quantum information processing, like strongly modulating pulses, are adopted to the quadrupolar spin system to implement the unitary operations and to generate pseudo pure states. Experimentally a set of quantum algorithms as well as the decoherence process is characterized by quantum process tomography which is developed for the quadrupolar spin system.
A detailed analyses of the basic conditions of the experimental quantum process tomography procedure is presented, showing that even in such non-Markovian systems this method is valid.
NMR quantum computing with qubit systems represented by nuclear spins (I=12) in small molecules in liquids has led to the most successful experimental quantum information processors so far. We use the quadrupolar spin-32 sodium nuclei of a NaNO3 single crystal as a virtual two-qubit system. The large quadrupolar coupling in comparison with the environmental interactions and the usage of strongly modulating pulses allow us to manipulate the system fast enough and at the same time keeping the decoherence reasonably slow. The experimental challenge is to characterize the "calculation" behavior of the quantum processor by process tomography which is here adapted to the quadrupolar spin system. The results of a selection of quantum gates and algorithms are presented as well as a detailed analysis of experimental results.
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.
The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically deterministic), and decoherent (microscopically random) components. This analysis yielded a superoperator consisting of a unitary part that was strongly correlated with the theoretically expected unitary superoperator of the quantum Fourier transform, an overall attenuation consistent with decoherence, and a residual portion that was not completely positive-although complete positivity is required for any quantum operation. By comparison with the results of computer simulations, the lack of complete positivity was shown to be largely a consequence of the incoherent errors which occurred over the full quantum process tomography procedure. These simulations further showed that coherent, incoherent, and decoherent errors can often be identified by their distinctive effects on the spectrum of the overall superoperator. The gate fidelity of the experimentally determined superoperator was 0.64, while the correlation coefficient between experimentally determined superoperator and the simulated superoperator was 0.79; most of the discrepancies with the simulations could be explained by the cumulative effect of small errors in the single qubit gates.
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.
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.