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arXiv:quant-ph/0304058 · The Journal of Chemical Physics

Nuclear magnetic resonance implementation of the Deutsch–Jozsa algorithm using different initial states

Vladimir L. Ermakov, B. M. Fung

Published May 29, 2003• 8 citations
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Abstract

The Deutsch–Jozsa algorithm distinguishes constant functions from balanced functions with a single evaluation. In the first part of the work, we present simulations of the nuclear magnetic resonance application of the Deutsch–Jozsa algorithm to a 3-spin system for all possible balanced functions. Three different kinds of initial states are considered: a thermal state, a pseudopure state, and a pair (difference) of pseudopure states. Then, simulations of several balanced functions and the two constant functions of a 5-spin system are described. Finally, corresponding experimental spectra obtained by using a 16-frequency pulse to create an input equivalent to either a constant function or a balanced function are presented, and the results are compared with those obtained from computer simulations.

DOI
10.1063/1.1574802

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The Deutsch–Jozsa algorithm distinguishes constant functions from balanced functions with a single evaluation. In the first part of the work, we present simulations of the nuclear magnetic resonance application of the Deutsch–Jozsa algorithm to a 3-spin system for all possible balanced functions. Three different kinds of initial states are considered: a thermal state, a pseudopure state, and a pair (difference) of pseudopure states. Then, simulations of several balanced functions and the two constant functions of a 5-spin system are described. Finally, corresponding experimental spectra obtained by using a 16-frequency pulse to create an input equivalent to either a constant function or a balanced function are presented, and the results are compared with those obtained from computer simulations.
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.
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.
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.
Despite early experimental tests of the Deutsch-Jozsa (DJ) algorithm, there have been only a very few nontrivial balanced functions tested for register number n>3n>3n>3. In this paper, we experimentally demonstrate the DJ algorithm in four- and five-qubit homonuclear spin systems by the nuclear-magnetic-resonance technique, by which we encode the one function evaluation into a long shaped pulse with the application of the gradient ascent algorithm. Our work, dramatically reducing the accumulated errors due to gate imperfections and relaxation, demonstrates a better implementation of the DJ algorithm.
The optimized version of the Deutsch-Jozsa algorithm proposed by Collins et al. was implemented using the three 19F^{19}\mathrm{F}19F nuclear spins of 2,3,4-trifluoroaniline as qubits. To emulate the behavior of pure quantum-mechanical states pseudopure states of the ensemble were prepared prior to execution of the algorithm. Full tomography of the density matrix was employed to obtain detailed information about initial, intermediate, and final states. Information, thus obtained, was applied to optimize the pulse sequences used. It is shown that substantial improvement of the fidelity of the preparation may be achieved by compensating the effects caused by the different relaxation behavior of the different substates of the density matrix. All manipulations of the quantum states were performed under the conditions of unresolved spin-spin interactions.