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arXiv:quant-ph/0105045

Orchestrating an NMR quantum computation: the N=3 Deutsch-Jozsa algorithm

David C. Collins, K. W. Kim, William C. Holton, Hanna S. Gracz, Edward O. Stejskal

Published May 10, 2001• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0105045

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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.
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 computing using two-dimensional NMR has recently been described using scalar coupling evolution technique [J. Chem. Phys.,109,10603 (1998)]. In the present paper, we describe two-dimensional NMR quantum computing with the help of selective pulses. A number of logic gates are implemented using two and three qubits with one extra observer spin. Some many-in-one gates (or Portmanteau gates) are implemented. Toffoli gate (or AND/NAND gate) and OR/NOR gates are implemented on three qubits. Deutsch-Jozsa quantum algorithm for one and two qubits, using one extra work qubit, has also been implemented using selective pulses after creating a coherent superposition state, in the two-dimensional methodology.
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 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.
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.