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arXiv:quant-ph/0206106 · Physical Review A

Implementation schemes in NMR of quantum processors and the Deutsch-Jozsa algorithm by using virtual spin representation

A. R. Kessel, Natalia M. Yakovleva

Published Dec 31, 2002• 27 citations
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Abstract

Schemes of experimental realization of the main two-qubit processors for quantum computers and the Deutsch-Jozsa algorithm are derived in virtual spin representation. The results are applicable for every four quantum states allowing the required properties for quantum processor implementation if for qubit encoding, virtual spin representation is used. A four-dimensional Hilbert space of nuclear spin $3/2$ is considered in detail for this aim.

DOI
10.1103/physreva.66.062322

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Schemes of experimental realization of the main two-qubit processors for quantum computers and the Deutsch-Jozsa algorithm are derived in virtual spin representation. The results are applicable for every four quantum states allowing the required properties for quantum processor implementation if for qubit encoding, virtual spin representation is used. A four-dimensional Hilbert space of nuclear spin 3/23/23/2 is considered in detail for this aim.
Recent years, remarkable progresses in experimental realization of quantum information have been made, especially based on nuclear magnetic resonance (NMR) theory. In all quantum algorithms, Deutsch-Jozsa algorithm has been widely studied. It can be realized on NMR quantum computer and also can be simplified by using the Cirac's scheme. In this paper, at first the principle of Deutsch-Jozsa quantum algorithm is analyzed, then we implement the seven-qubit Deutsch-Jozsa algorithm on NMR quantum computer.
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.
Quantum Computation is slowly evolving as the next big innovation that would change the entire way we perceive the world around us. Even though many scientists are actively researching on it, one main hurdle is that a physical quantum computer is not readily available to everyone to work on. The present paper computationally simulates a quantum algorithm using nuclear magnetic resonance on four different kinds of molecules and shows that the spectra obtained are similar to the experimental results. Computational simulations like these would allow researchers to verify their quantum algorithms using NMR without actually having a physical device hence overcoming the hurdle.
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.
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.
Quantum adiabatic algorithms is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. The adiabatic Deutsch-Jozsa algorithm given be Saurya Das, Randy Kobes and Gabor Kunstatter [Phys. Rev. A. 65, 062301 (2002)], shows how the Deutsch-Jozsa algorithm can be implemented in N^{1/2} steps, where N=2^n and n is the number of qubits, using local adiabatic evolution. In this paper we report the experimental implementation of the adiabatic Deutsch-Jozsa algorithm in a two qubit quantum information processor by Nuclear Magnetic Resonance.