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arXiv:quant-ph/0510115 · Physical Review A

Solid-state NMR three-qubit homonuclear system for quantum-information processing: Control and characterization

Jonathan Baugh, Osama Moussa, Colm A. Ryan, Raymond Laflamme, Chandrasekhar Ramanathan, Timothy F. Havel, David G. Cory

Published Feb 2, 2006• 44 citations
arXivHTMLPDF

Abstract

A three-qubit $^{13}\mathrm{C}$ 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 $^{13}\mathrm{C}--^{13}\mathrm{C}$ 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.

DOI
10.1103/physreva.73.022305

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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.
The enormous theoretical potential of Quantum Information Processing (QIP) is driving the pursuit for its practical realization by various physical techniques. Currently Nuclear Magnetic Resonance (NMR) has been the forerunner by demonstrating a majority of quantum algorithms. In NMR, spin systems consisting of coupled nuclear spins are utilized as qubits. In order to carry out QIP, a spin system has to meet two major requirements: (i) qubit addressability and (ii) mutual coupling among the qubits. It has been demonstrated that the magnitude of the mutual coupling among qubits can be increased by orienting the spin-systems in a liquid crystal matrix and utilizing the residual dipolar couplings. While utilizing residual dipolar couplings may be useful to increase the number of qubits, nuclei of same species (homonuclei) might become strongly coupled. In strongly coupled spin-systems, spins loose their individual identity of being qubits. We propose that even such strongly coupled spin-systems can be used for QIP and the qubit-manipulation can be achieved by transition-selective pulses. We demonstrate experimental preparation of pseudopure states, creation of maximally entangled states, implementation logic gates and implementation of Deutsch-Jozsa (DJ) algorithm in strongly coupled 2,3 and 4 spin systems. The energy levels of the strongly coupled 3 and 4 spin systems were obtained by using a Z-COSY experiment.
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.
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.
The goal of this project was to improve our control over nuclear spins in the solid state, and to extend this control to coupled electron-nuclear spins. Our key results are summarized below. 1. We extended the control techniques developed in liquid state NMR quantum information processing to the control of a three-qubit solid state NMR quantum information processor. 2. To explore multi-body spin dynamics and their sensitivity to decoherence, we have measured the decay of NMR multiple quantum coherence intensities in the presence and absence of the dipolar Hamiltonian, in a cubic and a one-dimensional spin system. 3. We also studied the transport of polarization in the one-dimensional fluoroapatite system, both experimentally and theoretically, and created special states in which the polarization is localized to the ends of the chain. This enables both universal Quantum Computing and Quantum Simulation in 1-D systems. 4. We devised a novel scheme for electron-nuclear quantum information processing that exploits the anisotropic hyperfine coupling. This scheme enables universal control over a 1-electron, N-nuclear spin system, addressing only a single electron spin transition. Not having to address the nuclear spins directly significantly speeds up the control. We designed and fabricated a pulsed electron spin resonance spectrometer, along with a cryogenic probe which we used to experimentally implement this scheme on a single crystal sample of irradiated malonic acid. 5. We demonstrated the role of nuclear spin dipolar diffusion in dynamic nuclear polarization (DNP) experiments, in dielectric samples with abundant nuclear spins. We achieved a 29Si polarization of 8.3% at 66 GHz and 1.1 K in single crystal P-doped, the highest ever reported, using DNP.
Residual dipolar couplings can be used to increase the number of qubits for quantum information processing. We have used molecules containing 3, 5 and 8 spins oriented in a liquid crystal matrix, and exploited the residual dipolar coupling to demonstrate quantum information processing in them. Transition assignment is performed using HET-Z-COSY experiment and qubit addressability is achieved by transition selective pulses. It is expected that using this protocol higher qubits can be achieved. For the implementations reported in this work, evolution under the internal Hamiltonian was not explored. It is however interesting to investigate how effectively the evolution under internal Hamiltonian can be manipulated to implement quantum algorithms in these systems. Recently an approach has been reported in this direction, where a new method of preparing pseudopure states in oriented systems by exciting selected multiple quantum using evolution under effective dipolar Hamiltonian, has been demonstrated [24].
We report the implementation of a 3-qubit quantum error-correction code on a quantum information processor realized by the magnetic resonance of carbon nuclei in a single crystal of malonic acid. The code corrects for phase errors induced on the qubits due to imperfect decoupling of the magnetic environment represented by nearby spins, as well as unwanted evolution under the internal Hamiltonian. We also experimentally demonstrate sufficiently high-fidelity control to implement two rounds of quantum error correction. This is a demonstration of state-of-the-art control in solid state nuclear magnetic resonance, a leading test bed for the implementation of quantum algorithms.