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The Journal of Chemical Physics

Nuclear magnetic resonance quantum computing exploiting the pure spin state of para hydrogen

Patrick Hübler, Joachim Bargon, Steffen J. Glaser

Published Aug 8, 2000• 38 citations
Publisher

Abstract

Nuclear magnetic resonance (NMR) is well-suited to implement quantum algorithms experimentally. However, there are serious problems associated with the noisy mixed initial state that is described by the thermal equilibrium density operator of NMR spectroscopy. Here we present a new strategy to dramatically increase the sensitivity of a NMR quantum computing experiment. Para hydrogen can be used to prepare a density operator in a suitable molecule that is very close to a pure state, an improvement on the order of 104 compared to “conventional” NMR quantum computing. Our strategy is demonstrated experimentally solving the Deutsch–Jozsa problem based on para hydrogen and Vaska’s complex.

DOI
10.1063/1.482015

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arXiv:quant-ph/0312014•Physical Review Letters•Jul 19, 2004•82 citations
Nuclear magnetic resonance (NMR) is well-suited to implement quantum algorithms experimentally. However, there are serious problems associated with the noisy mixed initial state that is described by the thermal equilibrium density operator of NMR spectroscopy. Here we present a new strategy to dramatically increase the sensitivity of a NMR quantum computing experiment. Para hydrogen can be used to prepare a density operator in a suitable molecule that is very close to a pure state, an improvement on the order of 104 compared to “conventional” NMR quantum computing. Our strategy is demonstrated experimentally solving the Deutsch–Jozsa problem based on para hydrogen and Vaska’s complex.
While quantum computing offers the potential of using new quantum algorithms to tackle problems that are intractable for classical processors, its implementation requires the development of quantum devices, which are as yet unavailable. The most complex implementations of quantum algorithms to date have used techniques adapted from nuclear magnetic resonance (NMR) spectroscopy, but current liquid state NMR approaches cannot be extended to systems with many quantum bits, as it is not possible to prepare pure initial states by directly cooling the spin system into its ground state. Furthermore, it has been shown that current NMR experiments involve only separable states, and thus could in principle be described by local hidden variable models. Here we demonstrate how para-hydrogen can be used to prepare a two-spin system in an almost pure state and so overcome these two objections. To achieve polarisation comparable to our observed value of epsilon=0.916pm0.019\\epsilon = 0.916 \\pm 0.019epsilon=0.916pm0.019 by thermal means would require cooling to 6.4 mK in the 9.6 T field used, or a magnetic field of 0.45 MT at room temperature.
This thesis addresses the problems of initialization and separability in liquid state NMR based quantum information processors. We prepare pure quantum states lying above the entanglement threshold. Our pure state quantum computer derives its purity from the highly polarized nuclear spin states in the para-hydrogen molecule. The thesis begins with a critique of conventional NMR based quantum information processing outlining the major strengths and weaknesses of the technology. We describe the enhanced magnetic ordering of the nuclear spin states in para-hydrogen and an initialization experiment exploiting this effect to achieve pure, entangled states. These states can indeed be used as initial states in implementing quantum algorithms: we describe mplementations of the Deutsch and the Grover quantum algorithms. The "twirl" operation converts a completely arbitrary input state to a Werner singlet. The NMR implementation of this operation is taken up. We also analyze the possibility of sharing the purity of some highly polarized qubits in a quantum computer onto quantum subspaces of arbitrary dimensions, and whether these sharing operations increase or decrease the likelihood of entanglement.
An experimental implementation of an ensemble quantum algorithm is presented in which the thermal state is used as an initial state of the algorithm. More specifically, we use four active qubits to demonstrate the solution of the Deutsch-Jozsa problem using a nuclear magnetic resonance quantum computer. For the implemented algorithm the number of molecules in the NMR sample is irrelevant to the number of qubits available to an NMR quantum computer, and the initial state is chosen to be the state of thermal equilibrium, thereby avoiding the preparation of pseudopure states and the resulting exponential loss of signal as the number of qubits increases. As expected, measured spectra demonstrate a clear distinction between constant and balanced functions.
We offer an improved method for using a nuclear-magnetic-resonance quantum computer (NMRQC) to solve the Deutsch-Jozsa problem. Two known obstacles to the application of the NMRQC are an exponential diminishment of density-matrix elements with the number of bits, threatening weak signal levels; and the high cost of preparing a suitable starting state. A third obstacle is a heretofore unnoticed restriction on measurement operators available for use by a NMRQC. Variations on the function classes of the Deutsch-Jozsa problem are introduced, both to extend the range of problems advantageous for quantum computation and to escape all three obstacles to the use of a NMRQC. At the cost of an extra work bit and a polynomial increase in the number of gate operations required, the method solves the Deutsch-Jozsa problem while avoiding an exponential loss of the signal, preparation of a pseudopure state, and temporal averaging.
Nuclear magnetic resonance (NMR) has been widely used as a demonstrative medium for showcasing the ability for quantum computations to outperform classical ones. A large number of such experiments performed have been implementations of the Deutsch-Jozsa algorithm. It is known, however, that in some cases the Deutsch-Jozsa problem can be solved classically using as many queries to the black-box as in the quantum solution. In this paper we describe experiments in which we take the contrasting approach of using NMR as a classical computing medium, treating the nuclear spin vectors classically and utilising an alternative embedding of bits into the physical medium. This allows us to determine the actual Boolean function computed by the black-box for the n=1,2 cases, as opposed to only the nature (balanced or constant) as conventional quantum algorithms do. Discussion of these experiments leads to some clarification of the complications surrounding the comparison of different quantum algorithms, particularly black-box type algorithms.
Here we demonstrate how parahydrogen can be used to prepare a two-spin system in an almost pure state which is suitable for implementing nuclear magnetic resonance quantum computation. A 12 ns laser pulse is used to initiate a chemical reaction involving pure parahydrogen (the nuclear spin singlet of H2). The product, formed on the micros time scale, contains a hydrogen-derived two-spin system with an effective spin-state purity of 0.916. To achieve a comparable result by direct cooling would require an unmanageable (in the liquid state) temperature of 6.4 mK or an impractical magnetic field of 0.45 MT at room temperature. The resulting spin state has an entanglement of formation of 0.822 and cannot be described by local hidden variable models.