Quantum information processing by nuclear magnetic resonance on quadrupolar nuclei

J. Teles, E. R. DeAzevedo, J. C. C. Freitas, R. S. Sarthour, I. S. Oliveira, T. J. Bonagamba
2012 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
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
more » ... nt 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. Keywords: quantum information processing; quantum state tomography; nuclear magnetic resonance; quadrupolar nuclei on April 13, 2017 http://rsta.royalsocietypublishing.org/ Downloaded from Review. QIP by NMR on quadrupolar nuclei 4771 (PPS), radio frequency (RF) pulse design for logic gate implementation and reading procedures, e.g. quantum state tomography (QST), are handled very well, due mainly to the fine control of the quantum evolution of nuclear spins. Allied to these characteristics are the relatively long relaxation times of nuclear magnetization, which, in the language of QIP, implies that the decoherence times of the nuclear quantum states are lengthy enough to perform the desired unitary evolutions. Nuclei with spin greater than 1 2 under the quadrupolar interaction provide an alternative means of creating an N -qubit system [1] [2] [3] [4] [5] [6] . The main advantages are the use of shorter pulse sequences than those applied to the spin systems in isotropic liquids and the representation of many qubits by only one nuclear species. Naturally, the same high strength of the quadrupolar couplings that enables the use of shorter pulses in quadrupolar systems gives rise to shorter relaxation times, leading to stronger decoherence and dissipation effects. However, this difference in the relaxation of quadrupolar and spin 1 2 systems arises from the distinct nature of the spin-environment interaction; so these systems can be used as models to investigate the differences in the decoherence and dissipation of quantum systems, in which the system-environment interaction is described by distinct quantum channels. Another characteristic of nuclear spins with strong quadrupolar moments is the possibility of performing QIP by means of nuclear quadrupole resonance (NQR) without using an external magnetic field. As a result of all these possibilities, there are many studies exploring QIP concepts by NMR in quadrupolar systems. The purpose of this study is to review some of the work published in this area and to present new results showing the excellent control over quadrupolar systems achieved by similar RF pulse techniques. QST is an important tool for characterizing the various stages of a quantum algorithm implementation. The first QST proposal for NMR was put forward by Chuang et al. [7] and further improved by Long et al. [8]. In those works, the authors proposed a technique for QST in heteronuclear coupled spin 1 2 systems. In heteronuclear systems, non-selective RF pulses can act on each nucleus separately. Such pulses generate individual spin rotations which, by means of specific pulse combinations, make it possible to project all the components needed to expand the system's density matrix in the NMR measurement operator. In the case of homonuclear spin 1 2 and quadrupolar systems, nonselective pulses produce only global rotations of all qubits: it is not possible to rotate one qubit state at a time. Therefore, to address the specific qubit states of such systems, several approaches have been proposed to use selective RF pulses to excite specific nuclear transitions [2, 3, 5, 6, 9] . However, owing to the long duration of selective pulses, relative to the non-selective ones used in heteronuclear systems, relaxation effects can severely restrict the use of QST methods. This restriction turns out to be more important if the number of spins increases or the spin quantum number is greater than 1 2 , because in these cases, many more selective excitations are necessary to find all the elements of the density matrix [6] . As a way to overcome these limitations, we proposed a QST method using only short non-selective RF pulses with a coherence selection scheme [10] . As a result, many studies involving quadrupolar nuclei could benefit from that method, including research on quadrupolar spin decoherence and relaxation [11] [12] [13] [14] and quantum simulation [15] . Section 2 briefly expounds the main concepts regarding PPS and logic gate implementation and QST in high-field NMR of quadrupolar nuclei. To illustrate the fine control obtained, Phil. Trans. R. Soc. A (2012)
doi:10.1098/rsta.2011.0365 pmid:22946040 fatcat:fynmbyvxhrdunnlfpmw737mpjy