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"Spectral Implementation" for creating pseudo-pure states and a three-qubit Bernstein-Vazirani's algorithm in NMR

Xinhua Peng, Xiwen Zhu, Ximing Fang, Mang Feng, Maili Liu, Kelin Gao

Published Feb 1, 2002• 0 citations
OpenAlex

Abstract

Based on ''spectral implementation'' proposed by Madi et al. (J. Chem. Phys. 109, 10603, 1998), we have experimentally realized a pseudo-pure state by the line-selective excitation and a three-qubit Bernstein-Vazirani's algorithm using a carbon-13 analine sample in NMR. The superiority of quantum computation to classical counterpart is well displayed.

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Quantum computers are able to operate on coherent superpositions of states, and to isolate a single global property of the set of computed quantities via interference. In principle, this permits them to solve certain problems exponentially faster thana classical computer, but not one has yet succeeded in implementing a true quantum computer on more than two quantum bits. Recently, however, it has been found that an ensemble of independent and identical quantum computers can perform mots of the same feats than a single quantum computer could, while at the same time bringing massive classical parallelism to bear on its computations. Such an ensemble quantum computer can be realized, to a limited extent, by nuclear magnetic resonance (NMR) spectroscopy on ordinary liquids at room temperature and pressure. This simple implementation depends on special kinds of mixed states, called pseudo-pure states, whose preparation entails a loss of signal that is exponential in the number of spins. While this would appear to limit such an implementation to ca. 8-10 spins in the foreseeable future, NMR spectroscopy has now permitted the first experimental demonstrations of all the basic features of quantum computing. We claim, moreover, that the product operator formalism, on which the theory of NM R spectroscopy is based, provides an efficient framework within which to analyze algorithms and decoherence effects in quantum computing more generally. This is illustrated by presenting our recent experimentally implementation of a quantum error correcting code.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 6, 1998•0 citations
Based on ''spectral implementation'' proposed by Madi et al. (J. Chem. Phys. 109, 10603, 1998), we have experimentally realized a pseudo-pure state by the line-selective excitation and a three-qubit Bernstein-Vazirani's algorithm using a carbon-13 analine sample in NMR. The superiority of quantum computation to classical counterpart is well displayed.
A quantum circuit is introduced to describe the preparation of a labeled pseudo-pure state by multiplet-component excitation scheme which has been experimentally implemented on a 4-qubit nuclear magnetic resonance quantum processor. Meanwhile, we theoretically analyze and numerically investigate the low-power selective single-pulse implementation of a controlled-rotation gate, which manifests its validity in our experiment. Based on the labeled pseudo-pure state prepared, a 3-qubit Bernstein-Vazirani algorithm has been experimentally demonstrated by spectral implementation. The "answers" of the computations are identified from the split peak positions in the spectra of the observer spin, which are equivalent to projective measurements required by the algorithms.
A new method of preparing the pseudo-pure state of a spin system for quantum computation in liquid nuclear magnetic resonance (NMR) was put forward and demonstrated experimentally. Applying appropriately connected line-selective pulses simultaneously and a field gradient pulse techniques we acquired straightforwardly all pseudo-pure states for two qubits in a single experiment much efficiently. The signal intensity with the pseudo-pure state prepared in this way is the same as that of temporal averaging. Our method is suitable for the system with arbitrary numbers of qubits. As an example of application, a highly structured search algorithm—-Hogg’s algorithm was also performed on the pseudo-pure state | 00 〉 prepared by our method.
Quantum information processing has been effectively demonstrated on a small number of qubits by nuclear magnetic resonance. An important subroutine in any computing is the readout of the output. "Spectral implementation" originally suggested by Z. L. Madi, R. Bruschweiler, and R. R. Ernst [J. Chem. Phys. 109, 10603 (1999)], provides an elegant method of readout with the use of an extra "observer" qubit. At the end of computation, detection of the observer qubit provides the output via the multiplet structure of its spectrum. In spectral implementation by two-dimensional experiment the observer qubit retains the memory of input state during computation, thereby providing correlated information on input and output, in the same spectrum. Spectral implementation of Grover's search algorithm, approximate quantum counting, a modified version of Berstein-Vazirani problem, and Hogg's algorithm are demonstrated here in three- and four-qubit systems.
This article reviews recent work done by us an some initial steps towards the implementation of quantum computation using liquid state NMR. We describe how special kinds of states required for such computation (called pseudo-pure states) can be created from a thermal ensemble of spins. We demonstrate the implementation of several quantum logic gates through one- and two-dimensional NMR methods, using transition- and spin- selective pulses. Finally, we discuss the implementation of the Deutsch-Jozsa algorithm using NMR.
A new approach to the implementation of a quantum computer by high-resolution nuclear magnetic resonance (NMR) is described. The key feature is that two or more line-selective radio-frequency pulses are applied simultaneously. A three-qubit quantum computer has been investigated using the 400 MHz NMR spectrum of the three coupled protons in 2,3-dibromopropanoic acid. It has been employed to implement the Deutsch-Jozsa algorithm for distinguishing between constant and balanced functions. The extension to systems containing more coupled spins is straightforward and does not require a more protracted experiment.
Quantum computers are able to operate on coherent superpositions of states, and to isolate a single global property of the set of computed quantities via interference. In principle, this permits them to solve certain problems exponentially faster thana classical computer, but not one has yet succeeded in implementing a true quantum computer on more than two quantum bits. Recently, however, it has been found that an ensemble of independent and identical quantum computers can perform mots of the same feats than a single quantum computer could, while at the same time bringing massive classical parallelism to bear on its computations. Such an ensemble quantum computer can be realized, to a limited extent, by nuclear magnetic resonance (NMR) spectroscopy on ordinary liquids at room temperature and pressure. This simple implementation depends on special kinds of mixed states, called pseudo-pure states, whose preparation entails a loss of signal that is exponential in the number of spins. While this would appear to limit such an implementation to ca. 8-10 spins in the foreseeable future, NMR spectroscopy has now permitted the first experimental demonstrations of all the basic features of quantum computing. We claim, moreover, that the product operator formalism, on which the theory of NM R spectroscopy is based, provides an efficient framework within which to analyze algorithms and decoherence effects in quantum computing more generally. This is illustrated by presenting our recent experimentally implementation of a quantum error correcting code.