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Physical Review A

NMR tomography of the three-qubit Deutsch-Jozsa algorithm

Oliver Mangold, Andreas Heidebrecht, Michael Mehring

Published Oct 11, 2004• 21 citations
Publisher

Abstract

The optimized version of the Deutsch-Jozsa algorithm proposed by Collins et al. was implemented using the three $^{19}\mathrm{F}$ 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.

DOI
10.1103/physreva.70.042307

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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.
A detailed description of the development of a three qubit NMR realization of the Deutsch-Jozsa algorithm [Collins et.al., Phys. Rev. A 62, 022304 (2000)] is provided. The theoretical and experimental techniques used for the reduction of the algorithm's evolution steps into a sequence of NMR pulses are discussed at length. This includes the description of general pulse sequence compilation techniques, various schemes for indirectly coupled gate realizations, experimental pulse parameterization techniques and bookkeeping methods for pulse phases.
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.
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.
The enormous theoretical potential of Quantum Information Processing (QIP) is driving the pursuit for its practical realization by various physical techniques. Currently Nuclear Magnetic Resonance (NMR) has been the forerunner by demonstrating a majority of quantum algorithms. In NMR, spin systems consisting of coupled nuclear spins are utilized as qubits. In order to carry out QIP, a spin system has to meet two major requirements: (i) qubit addressability and (ii) mutual coupling among the qubits. It has been demonstrated that the magnitude of the mutual coupling among qubits can be increased by orienting the spin-systems in a liquid crystal matrix and utilizing the residual dipolar couplings. While utilizing residual dipolar couplings may be useful to increase the number of qubits, nuclei of same species (homonuclei) might become strongly coupled. In strongly coupled spin-systems, spins loose their individual identity of being qubits. We propose that even such strongly coupled spin-systems can be used for QIP and the qubit-manipulation can be achieved by transition-selective pulses. We demonstrate experimental preparation of pseudopure states, creation of maximally entangled states, implementation logic gates and implementation of Deutsch-Jozsa (DJ) algorithm in strongly coupled 2,3 and 4 spin systems. The energy levels of the strongly coupled 3 and 4 spin systems were obtained by using a Z-COSY experiment.
A scheme to execute an \\eta-bit Deutsch-Jozsa (DJ) algorithm using \\eta qubits has been implemented for up to three qubits on an NMR quantum computer. For the one- and the two-bit Deutsch problem, the qubits do not get entangled, and the NMR implementation is achieved without using spin-spin interactions. It is for the three-bit case, that the manipulation of entangled states becomes essential. The interactions through scalar J-couplings in NMR spin systems have been exploited to implement entangling transformations required for the three bit DJ algorithm.
A growing appreciation among researchers is the use of quadrupolar nuclei with spin >1∕2 as a suitable candidate for quantum information processing. Such systems have multiple qubits per nuclei and large quadrupolar couplings, resulting in well separated lines in the spectrum. Preparation of pseudopure states and implementation of logic gates have been demonstrated in these systems. Two-qubit quantum algorithms have also been implemented in spin-3∕2 systems. In this letter we report implementation of three-qubit Deutsch-Jozsa algorithm [Proc. R. Soc. London, Ser. A 693, 553 (1992)] in a spin-7∕2 system of Cs133 nucleus oriented in a liquid crystalline matrix.