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IEEE International Advance Computing Conference

Simulation of NMR implementation of Deutsch-Jozsa algorithm

Varun Chappidi, Soumya Ganguly

Published Feb 1, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1109/iadcc.2014.6779431

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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.
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.
Quantum computing shows great promise for the solution of many difficult problems, such as the simulation of quantum systems and the factorization of large numbers. While the theory of quantum computing is fairly well understood, it has proved difficult to implement quantum computers in real physical systems. It has recently been shown that nuclear magnetic resonance (NMR) can be used to implement small quantum computers using the spin states of nuclei in carefully chosen small molecules. Here we demonstrate the use of a NMR quantum computer based on the pyrimidine base cytosine, and the implementation of a quantum algorithm to solve Deutsch’s problem (distinguishing between constant and balanced functions). This is the first successful implementation of a quantum algorithm on any physical system.
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.
The simulation of the spectra measured in nuclear magnetic resonance (NMR) spectroscopy experiments is a computationally non-trivial problem which, due to its natural interpretation as a quantum spin problem, maps in a straightforward way to a quantum computer. As such, it represents a problem for which such a device may provide some practical advantage over traditional computing methods. In order to understand the extent to which such problems may indeed provide examples of useful quantum advantage, it is important to understand the limitations of existing classical simulation methods. In this work, we benchmark our own classical solver designed to study such problems. This solver uses a clustering approximation to achieve a resource scaling which is linear in the total number of nuclear spins in a given molecule, for a fixed cluster size. The success of such an approximation would present a stark repudiation to the common claim that such problems require an exponential scaling of resources, the very claim which makes simulating an NMR spectra a candidate for quantum advantage. Our benchmarking results indicate that our approximation performs well throughout, and even somewhat beyond, the more typical experimental regimes. We discuss what implications this may have for future efforts to demonstrate quantum advantage in the context of NMR.
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.
Nuclear-magnetic-resonance (NMR) quantum computation is a fruitful arena in which to develop and demonstrate an enhanced capability for quantum control over molecular systems, regardless of the prospects, which may be limited, for building a quantum computer superior to a conventional computer for any computing task. We demonstrate a five-bit NMR quantum computer that distinguishes among various functions on four bits, making use of quantum parallelism, an example of the Deutsch-Jozsa problem. Its construction draws on the recognition of the sufficiency of linear coupling along a chain of nuclear spins, the synthesis of a suitably coupled molecule containing four distinct nuclear species, and the use of a multichannel spectrometer. Radio-frequency pulse sequences are described to execute controlled-NOT gates on two adjoining spins while leaving the other three spins essentially unaffected.