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Physical Review A

Method of multifrequency excitation for creating pseudopure states for NMR quantum computing

Anatoly K. Khitrin, Hong Yue Sun, B. M. Fung

Published Jan 5, 2001• 56 citations
Publisher

Abstract

An alternative method of preparing pseudopure states for NMR quantum computing is presented. The method involves population transfer by the use of a multifrequency excitation pulse. The system studied was a spin-7/2 nucleus ${(}^{133}\mathrm{Cs})$ in a liquid-crystalline environment. It is shown that all eight pseudopure quantum states for this three-qubit system can be prepared by using only 1--3 pulses, while the intensities of the relevant peaks increase by a factor of up to 4 compared to the equilibrium NMR spectrum. A comparison of the intensity enhancement by population transfer for quadrupolar nuclei and that for coupled spin-1/2 systems is given.

DOI
10.1103/physreva.63.020301

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An alternative method of preparing pseudopure states for NMR quantum computing is presented. The method involves population transfer by the use of a multifrequency excitation pulse. The system studied was a spin-7/2 nucleus (133Cs){(}^{133}\mathrm{Cs})(133Cs) in a liquid-crystalline environment. It is shown that all eight pseudopure quantum states for this three-qubit system can be prepared by using only 1--3 pulses, while the intensities of the relevant peaks increase by a factor of up to 4 compared to the equilibrium NMR spectrum. A comparison of the intensity enhancement by population transfer for quadrupolar nuclei and that for coupled spin-1/2 systems is given.
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.
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.
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.
Theoretical Quantum Information Processing (QIP) has matured from the use of qubits to the use of qudits (systems having states >2). Whereas most of the experimental implementations have been performed using qubits, little experimental work has been carried out using qudits as yet. In this paper we demonstrate experimental realization of a qutrit system by nuclear magnetic resonance (NMR), utilizing deuterium (spin-1) nuclei partially oriented in liquid crystalline phase. Preparation of pseudopure states and implementation of unitary operations are demonstrated in this single-qutrit system, using transition selective pulses.
The use of pairs of pseudopure states (POPS) as the basis for nuclear magnetic resonance (NMR) quantum computing is a method recently introduced. The advantage of this approach is its experimental simplicity: each POPS can be prepared in a single experiment with two steps, the first step having two pulses and the second having only one. The POPS method is now demonstrated for a 4-qubit and a 5-qubit system, for which the F19 and H1 spectra are first order and all the peaks are completely resolved. The characteristics of the POPS spectra are examined in detail, and the spectra of one of the five available sets of POPS for the 5-qubit system are shown as an illustration. The result of applying a c4-NOT quantum logic gate (controlled-NOT gate conditional upon the state of four other qubits) to another set of POPS in the 5-qubit system is also shown.
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.