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The Journal of Chemical Physics

Three-qubit nuclear magnetic resonance quantum information processing with a single-crystal solid

Garett M. Leskowitz, Nima Ghaderi, Ryan A. Olsen, Leonard J. Mueller

Published Jul 3, 2003• 26 citations
Publisher

Abstract

We report the first implementation of NMR quantum information processing in a static single-crystal solid, illustrating pseudopure state preparation and gate manipulations on a three-qubit system, U–13C2, N15-glycine (H215N13CH213COOH). We consider issues important to single-crystal NMR implementations of quantum information processing and introduce several innovations specific to such implementations. In particular, concurrent cross-polarization of both the C13 and N15 spins from proton magnetization enhances and equalizes populations on these nuclei in a single step. This multiple cross-polarization step simplifies preparation of pseudopure spin states relative to previously described multiple-pulse sequences. Also, proton decoupling, which prolongs coherence during spin-evolution and detection periods, is switched off to dephase unwanted coherences during preparation of pseudopure spin states. Such “relaxation averaging” (or T2 averaging) is an alternative to spatial averaging, which requires special hardware and large gradient strengths for small crystals.

DOI
10.1063/1.1582171

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We report the first implementation of NMR quantum information processing in a static single-crystal solid, illustrating pseudopure state preparation and gate manipulations on a three-qubit system, U–13C2, N15-glycine (H215N13CH213COOH). We consider issues important to single-crystal NMR implementations of quantum information processing and introduce several innovations specific to such implementations. In particular, concurrent cross-polarization of both the C13 and N15 spins from proton magnetization enhances and equalizes populations on these nuclei in a single step. This multiple cross-polarization step simplifies preparation of pseudopure spin states relative to previously described multiple-pulse sequences. Also, proton decoupling, which prolongs coherence during spin-evolution and detection periods, is switched off to dephase unwanted coherences during preparation of pseudopure spin states. Such “relaxation averaging” (or T2 averaging) is an alternative to spatial averaging, which requires special hardware and large gradient strengths for small crystals.
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.
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.
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.
The use of nuclear magnetic resonance (NMR) to carry out quantum information processing (QIP) often requires the preparation, transformation, and detection of pseudopure states. In our previous work, it was shown that the use of pairs of pseudopure states (POPS) as a basis for QIP is very convenient because of the simplicity in experimental execution. It is now further demonstrated that the product of the NMR spectra corresponding to two sets of POPS that share a common pseudopure state has the same peak frequencies as those of the common (single) pseudopure state. Examples of applying two different quantum logic gates to a 5-qubit system are given.
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The premise of nuclear magnetic resonance (NMR) quantum computing is that pseudopure quantum states, for which the populations of all energy levels but one are equal, can be used instead of pure quantum states. However, for systems with a large number of quantum bits (qubits), it would require elaborate pulse sequences with many steps to prepare these pseudopure states. To overcome this problem, a method using pairs of pseudopure states (POPS) instead of individual pseudopure states as the basis for NMR quantum computing is introduced here. The POPS approach has a significant advantage over the existing methods that, regardless of the number of qubits, only a single step is needed to prepare each pair of pseudopure states. A detailed analysis of a 16-level spin system is given, and the results of applying a c2{c}^{2}c2-NOT quantum logic gate (a controlled-NOT gate conditional upon the state of two other qubits) to all the POPS in the system are presented.