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Science

This Week in Science

Published Jan 17, 1997• 0 citations
Publisher

Abstract

Certain difficult computational problems, such as finding prime factors, could be performed much faster if quantum computing, in which interactions of pure quantum states correspond to logical operations, could be realized in a practical manner. Most schemes implemented or proposed require isolating or controlling single atoms to maintain the coherence of quantum states. Gershenfeld and Chuang (p. 350; see the news story by Taubes, p. 307) propose a different approach in which the manipulation of nuclear spins in a chemical compound with nuclear magnetic resonance (NMR) methods performs quantum computation. Such an approach might seem improbable in that the nuclear spins of a bulk sample form a statistical ensemble that would rapidly undergo decoherence. However, the authors show that the deviation from the equilibrium spin states can be manipulated so that it acts like a pure, low-dimensional quantum state.

DOI
10.1126/science.1997.275.5298.twis

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Certain difficult computational problems, such as finding prime factors, could be performed much faster if quantum computing, in which interactions of pure quantum states correspond to logical operations, could be realized in a practical manner. Most schemes implemented or proposed require isolating or controlling single atoms to maintain the coherence of quantum states. Gershenfeld and Chuang (p. 350; see the news story by Taubes, p. 307) propose a different approach in which the manipulation of nuclear spins in a chemical compound with nuclear magnetic resonance (NMR) methods performs quantum computation. Such an approach might seem improbable in that the nuclear spins of a bulk sample form a statistical ensemble that would rapidly undergo decoherence. However, the authors show that the deviation from the equilibrium spin states can be manipulated so that it acts like a pure, low-dimensional quantum state.
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.
Quantum computation remains an enormously appealing but elusive goal. It is appealing because of its potential to perform superfast algorithms, such as finding prime factors in polynomial time, but also elusive because of the difficulty of simultaneously manipulating quantum degrees of freedom while preventing environmentally induced decoherence. A new approach to quantum computing is introduced based on the use of multiple-pulse resonance techniques to manipulate the small deviation from equilibrium of the density matrix of a macroscopic ensemble so that it appears to be the density matrix of a much lower dimensional pure state. A complete prescription for quantum computing is given for such a system.
We show that quantum computation is possible with mixed states instead of pure states as inputs. This is performed by embedding within the mixed state a subspace that transforms like a pure state and that can be identified by labelling it based on logical (spin), temporal, or spatial degrees of freedom. This permits quantum computation to be realized with bulk ensembles far from the ground state. Experimental results are presented for quantum gates and circuits implemented with liquid nuclear magnetic resonance techniques and verified by quantum state tomography.
In bulk quantum computation one can manipulate a large number of indistinguishable quantum computers by parallel unitary operations and measure expectation values of certain observables with limited sensitivity. The initial state of each computer in the ensemble is known but not pure. Methods for obtaining effective pure input states by a series of manipulations have been described by Gershenfeld and Chuang (logical labeling) [Science 275, 350 (1997)] and Cory et al. (spatial averaging) [Proc. Natl. Acad. Sci. USA 94, 1634 (1997)] for the case of quantum computation with nuclear magnetic resonance. We give a different technique called temporal averaging. This method is based on classical randomization, requires no ancilla quantum bits, and can be implemented in nuclear magnetic resonance without using gradient fields. We introduce several temporal averaging algorithms suitable for both high-temperature and low-temperature bulk quantum computing and analyze the signal-to-noise behavior of each. Most of these algorithms require only a constant multiple of the number of experiments needed by the other methods for creating effective pure states.
Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor's quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. Rather than pursuing more complex algorithms on larger systems, interest has now largely moved into developing techniques for the precise and efficient manipulation of spin states with the aim of developing methods that can be applied in other more scalable technologies and within conventional NMR. However, the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.
If a computer could be built from quantum elements, which can exist in superpositions of many states rather than the 1s and 0s of conventional binary logic, then important hard problems could be solved rapidly. One example is the prime number factorization problem at the heart of modern encryption: given the product of two large primes, find the primes themselves. Another important problem is exhaustive search: given an unstructured list of items, quickly find the one that satisfies a certain property. A pair of Research Commentaries in this issue discuss some important new results in which simple quantum computers have been constructed with liquids probed by nuclear magnetic resonance techniques. The first, by L. K. Grover of Bell Labs, describes the theoretical situation and the implementation of an algorithm that he designed for exhaustive search. The second, by J. A. Jones of Oxford University, UK, discusses the recent experimental approaches to quantum computing with nuclear magnetic resonance.