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Fourier 1-norm and quantum speed-up

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Abstract

Understanding quantum speed-up over classical computing is fundamental for the development of efficient quantum algorithms. In this paper, we study such problem within the framework of the quantum query model, which represents the probability of output \(x \in \{0,1\}^n\) as a function \(\pi (x)\). We present a classical simulation for output probabilities \(\pi \), whose error depends on the Fourier 1-norm of \(\pi \). Such dependence implies upper-bounds for the quotient between the number of queries applied by an optimal classical algorithm and our quantum algorithm, respectively. These upper-bounds show a strong relation between Fourier 1-norm and quantum parallelism. We show applications to query complexity.

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Notes

  1. For details and up-to-date examples, the reader may refer to S. Jordan, Quantum Algorithm Zoo, https://math.nist.gov/quantum/zoo/.

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Acknowledgements

The authors thank R. Portugal, S. Collier, J. Szwarcfiter, and E. Galvão for useful discussions and suggestions. This work was initiated while SAG was at the Federal University of Rio de Janeiro, Brazil. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro - Brasil (FAPERJ), JCNE Grant E-26/203.235/2016.

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Authors and Affiliations

  1. Universidad Autónoma de Asunción, Jejuí 667, Asunción, Brazil

    Sebastián Alberto Grillo

  2. Universidade Federal do Rio de Janeiro, Avenida Horacio Macedo, 2030, COPPE/H319, Rio de Janeiro, Brazil

    Franklin de Lima Marquezino

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  1. Sebastián Alberto Grillo
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  2. Franklin de Lima Marquezino
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Correspondence to Franklin de Lima Marquezino.

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Grillo, S.A., de Lima Marquezino, F. Fourier 1-norm and quantum speed-up. Quantum Inf Process 18, 99 (2019). https://doi.org/10.1007/s11128-019-2208-7

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